MTG

Creatures do count as existing when their ETBs resolve. Generous Patron and similar triggers do count themselves. They count as having entered even before the ETB goes on the stack - see [[Aether Adept]]'s ruling - it can target itself, and targets for abilities are chosen before an ability is put on the stack.

Triggered abilities always go on the stack anyway - even if Generous Patron didn't count as existing when the ability was triggered (which it does), resolving the next action on the stack requires the current stack action to finish resolving, so it would have to be fully on the battlefield for any abilities triggered by it to be processed.

Your Generous Patron example is fairly simple. I think you came to the wrong conclusion when you say "the creature doesn’t yet exist when the ETB resolves." The creature definitely does exist at that point -- its existence is what triggered the ability.

If you want to use Generous Patron to put +1/+1 counters on creatures you don't control, and thereby get to draw cards from it, you can. Generous Patron has to be on the battlefield in order for its Support ability to trigger. If you choose to target other players' creatures with it, and if the Patron doesn't get killed before the Support ability resolves, it will be there to see you put those +1/+1 counters on things.

If you look at the rulings for Patron in Gatherer or Scryfall, you'll notice they point out that "Support can target a creature another player controls." That's trying to draw your attention to the fact that you can use Patron in this way.

Let's dig a little deeper. As a general rule, can entering creatures trigger their own ETB abilities?

Usually creatures will use specific wording to avoid confusion. They might say "Whenever another creature enters", like [[Celebrity Fencer]]. Or else they'll say "Whenever [this creature] or another creature enters", like [[Fallaji Vanguard]].

If you look at the (short) list of creatures that just say "whenever a creature enters", I don't think any of them meet the conditions to trigger themselves. At least, not ordinarily. But what if, let's say, [[Sigil Captain]] enters the battlefield while [[Godhead of Awe]] is already in play? Does it give itself the counters?

I think it does. I found this in the Comprehensive Rules:

603.10. Normally, objects that exist immediately after an event are checked to see if the event matched any trigger conditions, and continuous effects that exist at that time are used to determine what the trigger conditions are and what the objects involved in the event look like. [...]

So in my example, the "event" is Sigil Captain entering the battlefield, Sigil Captain is an "object that exists immediately after the event", and Godhead's second ability is a "continuous effect that exists at the time".

This answer was a lot longer than I thought it would be. I hope it makes sense. (I also hope it's correct.)