You were right!
In that type of situation, everyone is ready to begin fighting at any moment. Everyone basically is 'readying' for their fight - not just the PCs.
Readied Actions
These actions do require a trigger to be specifically stated, as well as the action they will take if triggered. In a general situation, most Readied Actions take place during combat when the turn structure is in place. However, it may be possible to ready an action outside of combat, but you as the DM will need to adjudicate the possibility of that.
From the Players Handbook, page 193 (emphasis mine)
...you can take the Ready action on your turn so that you can act later in the round using your reaction.
First, you decide what perceivable circumstance will trigger your reaction. Then you choose the action you will take in response to that trigger, or you choose to move up to your speed in response to it...When the tirgger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.
Your situation
In order for the players in your situation to their ready action, they would need to have explicitly stated what their trigger was and what the use of their reaction would be. However, given that the combatants were also likely 'readying actions', the initiative roll works because everyone is on even ground.
Even if this was a one-sided readied action event (where the combatants whom the PCs were fighting were not prepared for a fight), the trigger for the PCs readied action is the attack - which needs to be completed before they can act.
In addition, a readied action is just that, something that is being held waiting for the trigger. That could be a caster having started their spell and waiting for the event to finish casting it (if no event, the spell slot is still lost), or the bowman with their arrow nocked and bowstring pulled and ready to fire, etc. etc. That sort of action may also move towards a specific resolution (FIGHT!) rather than hoping for an alternate resolution (NEGOTIATE!)
The developers have your back
Jeremy Crawford Tweeted about when Readied Actions are available.
The options, including Ready, in the "Actions in Combat" section (PH, 192–93) are meant to be used in combat, after rolling initiative.
but he also suggested an alternative:
Your readiness can guard against being surprised. Otherwise, you roll initiative as normal. The DM might give advantage
Jeremy also had a nice discussion on this podcast about readied action and initiative (starts at about 6:10).
Unspecified Initiative Rolling
There is a potential issue in how you roll initiative that should be mentioned. This section is not a judgement against the actions described, but merely a reminder to have a consistent and agreed upon method of initiative at your table.
If you are trying to create a level playing field, then you should roll specifically and openly for each combatant. While the DM does have "control" to make changes behind the scenes, what you have done is to allow the situation to play out as you wanted it to rather than be dictated by the rolls of the dice.
Even though you had one guy ready to go, you could have created a narrative around why the other mook got to go first - and that would have been a more consistent and fair way to adjudicate your initiative rolls.
The decision to assign the highest roll to the guy who was about to shoot may have further incensed the players and made them feel they didn't get the opportunity they should have. It doesn't mean what you did was wrong, but it may have contributed the feelings at the table. It may not have, but I think it's something you should be aware of and consider when rolling your initiative.
In this scenario, your bandits were no doubt taken by surprise when one of them got shot. Unless there's some sort of telepathy or pre-planning involved, I would've ruled that the other PCs were also surprised.
In this case, for the first round all combatants bar your shifty PC are surprised (which is sometimes helpful to think of as a condition), meaning they miss their turn. During that turn, Mr Shifty shoots one of the bandits with his crossbow. At the end of the round, No one is Surprised and we follow on with normal combat. Thus according to my ruling, your PC would get one extra action. That's not that big a deal really, and your decision to just have Mr Shifty act first, out of turn, was reasonable.
Best Answer
Intent is not execution
I think the issue here is a disconnect between intent and execution. You might say "I'm done with this, Reynald pulls a dagger and stabs him, sneak attack." But that's not the execution of the action, that's only a declaration of what your character is going to try to do, and the dice rolls are going to determine whether it actually happens the way you said.
So how does that square with the enemy getting initiative and going first? Actually, it's simple: the shift from "argument" to "attempted murder" isn't necessarily caused by a dagger sliding between his ribs. Instead, that change in intent is as subtle or as telegraphed as the dice indicate. If the target wins initiative and gets to react before you attack, that's not reversing causality to react before the trigger; rather, that means something happened that tipped your hand too soon (or your opponent is just that fast).
You sort of skimmed past stating that the enemy isn't surprised, but that's a key point. If the transition from hostile argument to violence went unnoticed by the opponent, then you would have a surprise round, and you'd get your first strike even if your initiative was lagging. The fact that you didn't get a surprise round tells me that either something happened that let the other guy know you were going for your blade, or things were so tense already that he was prepared for you to throw down at any moment, and in either case that starts to explain the situation.
In any case, when the enemy wins initiative over you after you declare a sudden attack, it could mean your PC pulled a knife, but the target saw it before you could come in for the stab and got their weapon out faster (whether you were trying to be subtle and didn't realize you got spotted, or you were trying for speed and the target is just quicker than you, like a spaghetti western shootout). It could mean they saw your face change, or just sensed the surge of murderous intent like the hero in a Kung Fu movie, and knew instinctively that you had just decided to kill. It might mean that while your intent was an ambush, your character lost their cool and screamed, "THIS IS FOR MY FATHER!", which ruined the surprise.
As with many scenarios in D&D, the dice are telling you what actually happens, and it's up to your creativity to explain how that came about.