Introduction and Context
Over the course of almost two decades of gaming, and most of my real life, my experience is that there's a social approach which seems like one of the many workable choices 'on paper', but turns out to cause more problems than it solves in practice. I'm talking about the approach that involves getting what the PC(s) want(s) through threatening the NPCs.
Whenever I examine scenes in which such an approach either was or could be used, an overwhelming majority of the time I get the impression that it's not the best choice. That at best it'll provide a short-term gain at the cost of a greater long-term loss, which adds up to be a net negative throughout the whole length of a campaign.
Now, an overwhelming majority of systems which distinguish approaches, be they Rapport vs. Provoke, Charm vs. Intimidate, Respected vs. Feared, Persuasion vs. Intimidation, or whatever else they may call them, seem to not consider one of the two to be sufficiently inferior to warrant a discount or a disadvantage compensation. Similarly, I see that many players and GMs are much more comfortable with this approach and see it as quite worthwhile (e.g. willing to play a PC whose only way of socially dealing with others is through terror).
Based on that, I'm assuming that I'm just not very knowledgeable in the strategies of choosing social approaches, or am lacking some other knowledge or practical understanding – thus I'm asking how to mitigate the drawbacks and emphasise the benefits of intimidation-oriented approaches, keeping them 'worth it' as compared to other (primarily more 'positive') approaches to social interactions over the course of a campaign, and prevent them from turning into a downward spiral.
The Off-Putting Drawbacks
There are two key drawbacks that make me see such an approach as generally inferior, and that I want to learn to roleplay 'around' – either by roleplaying in a way that would reduce the disruptive effects thereof, or in a way that would make the drawback worth the benefit.
The primary drawback I see is that of consequences spanning beyond the short term. Even assuming the browbeating attempt is successful, it creates resentment and worsens future relationships with the NPC in question. Even if a power asymmetry makes the target unable to just avoid the PC in the future, this still creates risks down the line – this a contributor to why, for example, minions and dragons betray their evil overlords and switch teams at the most inconvenient moment. This isn't just a matter of tropes and fiction – based on the contrasting experience with workplaces where bosses use different social approaches, the ones that went for intimidation had a major long-term deterioration of the underlings' attitude and reduction of morale. It seems like if I play my PC as a thug, this is prone to creating a trail of enemies – enemies who either hate my PC for the actions in question, or even the whole party for being in cahoots with the thuggish jerk. (Can substitute another fearsome archetype in place of thuggish if you like.)
The secondary drawback is that of applicability and appropriateness. There are more situations where being likeable is useful than situations when being scary is useful. E.g. even in warfare, the grudging respect of the opponents tends to be of use. Conversely, there are many situations where being scary either doesn't confer worthwhile benefits, or has additional drawbacks (such as reactions of onlookers). Perhaps I'm just terrible at seeing the right moments for playing up the scariness of a PC in a useful way.
A Word on 'Naturally Occurring Situations'
The above observations are applied to what I call Naturally Occurring Situations in the context of a campaign. By that I mean that the GM tends to generate situations the way they would be logical for the setting and the campaign, without significantly bending the logic to cater to specific character builds. I understand that it's possible to deliberately come up with situations that would emphasise a specific approach, but I found most such approaches too contrived and thus not good for willing suspension of disbelief.
This also incidentally means that a large but not total part of my interest focuses on finding enough good uses of intimidation as a player or as a GM controlling a proactive NPC that would make it competitive with diplomatic/persuasive/charming approaches (which tend to be a much more universal tool in the kit, and much less prone to backfiring), when analysing situations that already came up. Conversely, answers that amount forcing the events to 'conspire' to raise menacing approach to usefulness/appropriateness level of the more likeable ones (whether through GM fiat or metanarrative influence given to players in narrative-leaning systems) are not helpful.
A Word on System Agnosticism
This was never intended to be a mechanical question; it's a question about roleplaying in a non-self-detrimental way. The observations of mine are gathered across playing using multiple systems, including campaigns which only nominally adhered to a system (being played very rules-light most of the time), or which use roleplaying, not mechanics, to resolve social situations. Thus the question is tagged as system agnostic. If some sort of non-agnostic default assumption cannot be abstained from, it's probably best to take 'freeform roleplay over IRC or other chat' as one for a starting point.
Best Answer
Intimidation Works
If you are looking for mitigations to the negative impact of intimidation-based approaches, but aren't talking about specific mechanical drawbacks, then really you just need to have a better understanding of how intimidation-based approaches work in the real world to run a PC that uses intimidation effectively.
Kinds Of Intimidation
First of all, intimidation isn't just threat of violence. You can divide "persuasion based approaches" as basically pushing the human "desire" button (whether lust, greed, or virtue) from "intimidation based approaches" as basically pushing the "fear" button - but it can be fear of losing your job, or of shame or humiliation, or of someone making a scene, or a bad performance review, or loss of some other item or privileges, or social troubles...
There are well documented social systems that work predominantly on intimidation that you can watch every day on TV, from Mean Girls to the Trump White House to the crime families in the Sopranos, to police interrogation, to, frankly, some parents. I mean, we wish, and are taught in elementary school, that approach isn't self sustaining, but it can be if you just double down on it enough.
What is the character using as leverage?
Now if a PC is just relying on "violence" type interrogation, that's like only relying on persuasion approaches where you offer up your body. It'll work some of the time, but not all of the time. You should be looking for any of the many kinds of undesirable things you can use as leverage in intimidation. Now, the more institutional power the PC has, the more options they have. If you're a murderhobo that just rode into town, then violence or embarrassment (like the Blues Brothers scene in the fancy restaurant) are your only options. The GM is never required to just "let someone roll" and push their skill button if they don't have any relevant leverage in any game I know of - for intimidation or persuasion. But even wandering PCs end up poking into things and uncovering secrets, etc. that can be used. "You wouldn't want the king to find out about this letter you wrote to the count!" The more prep work you do, the more (and safer) opportunities to use intimidation you'll find.
Intimidation Blowback?
Some assume this worsens relationships and makes NPCs less cooperative in the future. But is that really how the world works? They're less likely to do helpful stuff for you in your absence, sure. But in my experience, people who have caved in once to an intimidation are pretty much automatically likely to cave in to that same person next time even without the full threat machine having to come into play. The question states one down-side as the PC's friends are "looked down on" for associating with the thug which is true - but they are also implicitly other people to fear. "If I call the guard on him that's all well and good but what about those other four people with bloodstained clothing he rode in with?" The target doesn't have to know they're goody-goody, feel free and tell them "we'll all be watching."
It's up to your PC to develop a longitudinal approach to intimidation. If they just want to flex their pecs and wander off, then sure, it's not going to last long and there'll be blowback. Ever seen a TV/movie scene where the criminals (or cops, to be fair) initially intimidate someone and then ask for their drivers' license and get their address and say "you don't want to hear from us again?" That's a secondary intimidation designed to make them not just turn on them the second they walk out of frame. In The Sopranos, once you pay the gang protection money, then suddenly you have a rich set of potential problems - them messing with you, another crime organization messing with you, the cops messing with you, social stigma from dealing with criminals... All of which are leveraged to make a self-sustaining relationship based on intimidation.
Of course the Sopranos usually mix the greed in with the fear - mixed in are opportunities for people to make more money, etc. - using both approaches together is effective. But you don't have to rely on the skill...
Effective Intimidation In Game
I'm running a Pathfinder campaign right now where the PCs are pirates and the captain is Lawful Evil. He has certainly used physical intimidation on crew members, but also the threat of undesirable duties, the threat of withholding money, the threat of withholding shore leave, and so on. He also uses positive reinforcement to solidify his position, but he sucks at Diplomacy so he doesn't try to "persuade." Persuading says "Hey, if you go do this special mission for us I'll give you 500 gold!" He says, "Here's 500 gold. You're going on this mission. If you f**k it up you will regret it." Same parameters, but an intimidation based approach. While you're being technically intimidated into doing it, you're also up 500 gold at the end... Maybe you'll stay on this train and see where it's going.
Similarly, when threatening others, he doesn't just have to bare his teeth at people. "It'd be a shame for..." something to happen to someone else you care about, or your property, or for your business rival to get a windfall, or people to find out about that whorehouse you go to, or... Now, many of these aren't drive-by approaches, you need some intel (source of leverage) on the target first to make it long lasting, but the same bar should exist with persuasion: I "like" plenty of people but am not going to let them into my place of work without something special going on.
Conclusion
People don't like being intimidated, but with judicious application of carrot and stick, you can make a career of it.
So in the end, I am challenging your two assumptions about the drawbacks of intimidation based approaches, as they don't bear out in the real world as always bringing everything crashing down. Bullies do pretty well for themselves in many cases. You can't just do it dumb, but you shouldn't let people succeed at persuasion dumb either - similarly, that works for a while and then collapses (hint: Fyre Festival). @Glazius' answer is good in highlighting how to not overpower persuasion, but I've stuck mostly to the intimidation side of the equation.