I think I read somewhere that the designers of DND 5e didn’t like delaying actions, because this could screw with the duration of effects that last “until your next turn”. Effects like that would just last until your original place in the initiative order, not until your delayed turn.

This is going to be a fantastic way to get rid of paralysis effects or other major debuffs, or buffs on your target.

I delay until the monster's turn ends.

This is probably the exact case the designers were worried about. You can use the delaying system to easily wait until short-duration effects run out in the turn order.

Also, you have to think about what happens to things that let you save against them at the end of your turn; if you get to roll a new saving throw against Hold Person you could just delay your turn, roll the save at the end of your turn, then take your delayed turn without the downsides of the spell.

For more information; 4th edition had both "end-of-turn" effects and the Delay action. You might want to check how they handled it there, but it became quite messy over time due to exactly the kind of reasons that made the 5e designers decide to drop the feature.

Yes

Ready action requires (emphasis mine):

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.

The general rule is 1 Action per turn, but then specific rules can override that (e.g. Action Surge, Haste, etc.). I believe that Ready is another specific rule that overrides this as well because while you are using your Action to Ready, what you are really doing is using your Reaction to move your Action to another time.

Attack vs. attack (or does action have a meaning here?)

Upper case and lower case a have a meaning in the PHB, with the former being the Action described under Actions in Combat in the PHB.

When Ready asks you to choose your action, you use the actions listed there. The cases described in the PHB are "Use an Object" and "Movement" for the Ready (since you can only do one.)

In the case of this question, the Action that is being taken is Attack.

Attack Action on your turn

If you take the Attack action on your turn, then it comes with all the fun stuff that happen on your turn. This includes Extra Attack (which typically doesn't happen on a Ready whose trigger occurs NOT on your turn.)

In contrast, you can look at another reaction: Opportunity Attack. In this instance, the language does not say to take an action but to simply make a melee weapon attack. The difference is in the use of Action in the phrasing for Ready.

Actions vs attacks

The language in Ready is to choose the action. Actions are a defined term in the rules under Combat->Actions in Combat.

At other times, there is other language used when specifying melee/ranged/melee weapon/ranged weapon attacks. The language in Ready does not say to Ready an attack (lowercase a), it says to ready an action.

There are also several referenced examples of Ready Action and Extra Attack/Multiattack that reference the On your turn as the important qualifier. Had they not intended a Ready attack to be the Attack Action, then the discussion would have ended there and not referenced the turn requirements.

There are also other specific examples of more than one action on a turn that override that general rule: Cunning Action, Haste, Action Surge, etc.

Intent Support from Jeremy Crawford

The action choice intent (emphasis mine) is also provided by Crawford:

The Ready action lets you ready any action you can take, including Attack, but Extra Attack is on your turn.

Not only has he clearly stated that you take an Action, and that Action can be Attack (uppercase A), but that Extra Attack only functions on your turn.