The Real Problems Did Not Get Fixed
But let's talk about these individually.
Multiple Attribute Dependancy
Also known as MAD, this has plagued the poor Monk for ages. Monks need Strength for accuracy and damage, Dexterity and Wisdom for their armor class and two saving throws, Constitution for hit points (to be fair, all characters want this) and their last saving throw, Intelligence for skill points and Charisma to make use of their short list of social skills. This is not a good thing, because there's only so many high or even mid ability score numbers to go around. Pathfinder seemed like it was going to help this when it improved the skill system, consolidating many skills and relieving the reliance on Intelligence, but then it turned right back around and made boosting your ability scores with items more expensive, actually worsening this particular issue.
Anti-Synergistic Class Features
This just straight-up didn't change. Many of the Monk's class features are mutually exclusive with other features, with the use of feats, with the use of skills, or some delightfully awful combination of all three. Additionally, Monk still has problems making use of combat maneuvers (due both to low numbers and the increasing Size rating of stock enemies) and the feats relating to those maneuvers got weaker, cutting off a theoretical avenue of contribution.
More Melee, More Problems
Traditional problems with "mundane" or "melee" classes like Monk, but also like Fighter or Barbarian, were not solved by Pathfinder. They still have problems with enemies of all kinds that fly, burrow, teleport, cast spells, use "lockdown" effects like paralysis or poison, and utilize battlefield control (spiderwebs, choking fogs, etc) among other things. Like all melee classes, the Monk is forced to funnel enormous amounts of cash into meeting the increasing demand for complexity as levels and challenge ratings climb ever-higher. Unlike other melee classes, Monks cannot shore up their "primary" role, because...
Monks have no specialization
And, unlike Rogues or Bards, Monks cannot be made into competent generalists because of the aforementioned anti-synergy and low numbers, forcing them to pour resource after resource into badly mimicking another class's role.
On Monk's defence: the Archetype system
Pathfinder did introduce the Archetype system, a refinement on the idea of Alternate Class Features from 3.5e. Some Monk archetypes, such as Zen Archer and Hungry Ghost, work to alleviate some of these issues. Combinations of archetypes, done intelligently, may make for a playable character whose class still reads 'Monk'. Whether or not such a thing is worth the effort is not easily decided, but the options are available.
Monk changes very often
With that in mind, a word to the wise - Pathfinder gets errata often, and Monk has been the subject of many heated debates and quick rules changes, such as the brief-lived errata to Flurry of Blows. This does not have to affect your table, but if you're participating in a sanctioned table (like in Pathfinder Society) or if your group cares about such things your Monk may find the rules shifting out rapidly from underneath him. Caveat emptor.
Multiclassing at lower levels will be extremely powerful but progression later may feel stunted and less satisfying.
You said:
- The standard ability score improvement progression of 4th, 8th, 12th, 16th and 19th class level is based on character level instead. Additional ASIs such as the Fighter's are still attained at the listed class level.
Untethering ASI's from class level and giving them out at character levels 4, 8 (etc.) will incentivise multiclassing at lower levels for obvious reasons - but the flip side of that is it'll also discourage single class characters. At class level four (hit dice aside) the ASI is the only thing that numerous classes get (Barbarian, Fighter, Paladin, Ranger, Rogue).
With your suggested change a Fighter 3 / Monk 1 has, apart from an extra hit dice, all of the capabilities of a Fighter 4 / Monk 1. If they do decide to take Fighter 4 (if they want to access high level Fighter class features) they'll effectively get nothing except HP for doing so.
Similarly, you said:
- All classes that provide Extra Attack stack for the purposes of providing extra attacks up to the level they provide their last extra attack.
This actually compounds the issue further. Our Fighter 3 / Monk 1 already has his ASI. While he won't get his normal class benefit from Fighter 4 - if he does take it he would now get Extra Attack early instead - problem solved! Except the problem's just been deferred because taking Fighter 5 has been similarly devalued... and in fact it's even worse than that.
Why would our Fighter 3 / Monk 1 take Fighter 4 for extra attack, when he could take Monk 2 and get all of the 2nd level monk features AND extra attack as well? In the short term it would be inefficient not to take Monk 2 instead. But now you've reduced Fighter 4 and Fighter 5 to both being HP only. But it's actually even worse still!
With the next two fighter levels being useless it's likely you'd push forward in Monk at level 6 with Fighter 3 and Monk 3. Great. But now you won't get much by way of improvement from Monk 4 or Monk 5 either! Sure you'll get an ASI when you reach character level 8 - but you'll get that regardless so in the short term it'll probably feel like it's better to multiclass even further.
Note: When they take class levels that have been stripped of rewards, these characters won't actually be worse off than someone who multiclasses using RAW rather than your house rules. But giving multi-classed characters class features like ASI's and Extra Attack early to help them 'keep pace' with single class characters will make later progression feel like much more of a slog.
Having the cool features early will be fun (and is probably pretty unbalanced - but as you're not too bothered about party balance I won't go into that), later on though, it will feel like the characters have practically plateaued in their development as they gain very little on certain level choices. All of the above problems will be repeated when you hit class level 8, 12 (etc.).
Basically, levelling will, in general, feel much less even (rather than a relatively smooth curve) and you could back yourself into a corner with what feels like no good choices to choose from.
Best Answer
In short, yes.
There are a few things that make monk on par with the other melee fighting classes. I'll list a few of them here:
Dex to attack and damage
Monks can apply dex to attacks and damage automatically at level one.
Extra attacks
Monks are the only class that gets 2 attacks per round at level 1, and they keep up with Fighter attacks per round all the way to 20.
Class features apply to monk weapons
What counts as a monk weapon has been expanded a lot, and most of the monk class features (including increased damage) apply to monk weapons and unarmed strikes equally. In addition, the slew of exotic weapons that monks used to get have been condensed into reflavours of simple weapons. For example, nunchaku are now mechanically clubs, rather than a separate weapon type.
Less MAD
Monks in 5e have much less of a Multiple Attribute Dependency problem. Since monks get Dex to attack and damage for free, Strength is largely unnecessary. Int and Cha are as useful as they were in 3e. There are a few things that Int and Cha are useful for in both systems, but they're hardly essential.
Wisdom affects the same sort of stuff in 5e as it did in 3e, but it's less important due to other mechanical changes. Since "flat-footed" doesn't cause you to lose Dex Ac anymore, using Dex to increase AC is just as good as increasing Wis. You can probably end up with similar AC as a fighter by then end game.
Monks need to care about 3 stats, but they have basically the same number of dependencies as a fighter or barbarian. The fighter and barb both need Str and Con, with a secondary focus on Dex, and a monk needs Dex and Wis, with a secondary focus on Con. Since the stat cap is 20, it's actually not difficult to have a 20 and a couple 16-18s, as well as a 12 or two.
Better class features
The monk class features in 5e are similar to the ones in 3e, but they're a bit better, and a bit more reliable than they were before. I haven't done any real analysis on this, this is really just based on a first glance.
Interesting paths
There are three monk paths in the PHB, all of which do cool things. There's one for a classic monk that's similar to the one in 3e, but the other two add some really interesting things to the class that broaden the kind of situations that the monk can handle, which is a problem that mundane classes have had since the dawn of time. :)
Overview
Broadly speaking, the monk does less damage than the Barbarian, and has less tanking ability than the Fighter, but is a very strong and versatile mobile melee fighter. A better character optimizer might have a different opinion, but as a player who doesn't make broken characters, the monk looks like it's on par with the Barbarian and Fighter.