I didn't say it was all on luck. I said it was more on luck than it has been in the last two editions.
Fair enough.
How many rounds do Hold Person need to "stick" for the casting to be worthwhile?
If the spell is used to deny the opposition the target's actions, I would say three. If everybody can and will gang up on the target, perhaps two.
Thing is with the new edition the spell becomes better with level, since your save DC outpaces the monsters saves. At least if you cast intelligently, meaning at low-wis targets.
If I may exaggerate slightly:
At low levels, Hold Person has only a 50% chance of succeeding. But at mid- to high levels, this can shoot up to perhaps 75%. (Yes, casting it against level-appropriate foes)
The benefit is vastly more than 25%, since you make repeated saves. Let me show you.
The probability that a spell affects a target 1, 2 and 3 rounds are...
At 50% chance of saving: 50%, 25% and 12%.
At 25% chance of saving: 75%, 56% and 42%.
So while it is easy to think "75% is half as much more than 50%, so the spell is worth 1,5 times as much", the reality is that the spell is worth more than twice as much or even four times as much, depending on how many rounds it needs to stick for you to consider it.
So if I too advise the OP to wait and play more, this is based not so much on assuming it isn't used right, but more on how 5th Edition scales save DCs with your level, not the spell's level. And "bounded accuracy" visavi mediocre monster saves! Taken together, this can make a spell more valuable instead of less (as was the case in previous editions).
If you do make the spell more powerful (such as by your suggestion to not allow the save in round #1) you should probably also make an exception to the way save DCs increase with level. For instance, your variant always has a save DC 13 that never increases (from ability score or proficiency).