[RPG] Do legendary actions trigger effects that happen at the beginning of the monster’s turn

This is not a problem for most effects. Most of them last “until the end of your next turn”, which for effects that benefit something you want to do on your next turn is plenty long enough.

The only case where that produces counterintuitive results is when an effect disadvantages your opponent, as in your defensive spell example or a monk's Stunning Strike, because if your next turn comes before they next act then the effect is “wasted”.

This second class of effects is easy enough to identify on-the-fly though. Since you're already using a variant rule, it's easy (and totally true to the spirit of the rules as written) to house rule that such effects last “until the end the next round”, giving a full round plus a few initiative counts of effect, fully covering the opponent's next action.

The only drawback of that is that it might cover more than one of your opponent's next turns, if the initiative counts fall that way. Why do I say that's a drawback? Because the enemy can do the same to you, if they use such an effect against you and the dice go their way. However, that makes it pretty fair: the PCs might occasionally get the double-dip benefit, but so can the enemy. And the dice are likely to more often go in the heroes' favour, since that's the overall design of the rules.

Actions are not designed to happen at the same time, rather they happen in discrete chunks.

This is clear from the design of turn based games in general, and especially D&D. This is also apparent, as things that grant reactions usually specify specific timing, either before or after its trigger. Example of the Ready Action (PHB p. 193):

To do so, you can take the Ready action on your turn so that you can act later in the round using your reaction…[w]hen the trigger occurs, you can either take your reaction right after the trigger finishes or ignore the trigger.

There are no clearly written rules for resolving multiple readied actions on the same trigger.

However, the easiest, and most straightforward way to resolve this situation is in initiative order for all those readied on the same trigger, and seems to best fit the spirit of the rules, and how they work in general.

An alternative to this, would be to resolve order based on dexterity bonus, however, this introduces additional ties, which should already be resolved by initiative order.

Your last example does let the monster "save" for each attack, but it's still a valid and powerful tactic.

Preventing the monster from responding in between each attack is still a powerful benefit, and may potentially finish it off before it gets another turn.

Being hit while asleep from the Sleep spell, they would wake up on the first strike and lose the unconscious status, and being hit by something that gives a save on damage, such as Tasha's Hideous Laughter, would give them a save for each hit regardless of if all actions happen simultaneously or not, as per the wording of the spells, as each hit is still a separate instance of the target taking damage. Note that victims of Tasha's Hideous Laughter and Sleep would still be prone until they get a turn to stand up.

Finally, the general rule is that things are resolved in initiative order.

There is no specific rule overriding that. Following D&D 5e's system of general rules applying unless a specific one overrides it, I'd conclude this is likely the RAI and RAW approach. See SRD p. 90:

Initiative determines the order of turns during combat. [...] This is the order (called the initiative order) in which they act during each round. The initiative order remains the same from round to round.