"And here's a logistical question: Suppose you go to enter the subway and find you have insufficient fare left on your card? With a Metrocard, you go to the machine and add some fare. What will you do with your OMNY card?"
What I would do (and in fact will do) - put an amount on the OMNY Card that is equal to a specific number of rides (that you want to add) at the current fare. I do that now with my Reduced Fare MetroCard. And when it gets down to only a few fares I load more money on it. While I intend to use my RF MetroCard until it expires in May 2023, if I get my Reduced fare OMNY card earlier than that I'll just hold it and once I start to use it, if the machines have not be installed in stations I'll add money online as I need it.
[For the sake of transparency - I am a retired financial /budget analyst (worked for a large bank) so figuring these things out comes somewhat easy for me.]
If your OMNY card does run out of money (shame on you for not keeping track), if your credit or debit card is contactless you can always use it at the turnstile - you'll be charged the full fare rather than the reduced fare but at least you'll be able to enter.
If you want to, you can register the OMNY card,, link it to a debit or credit card and set up an auto refill - at an amount you specify when the balance gets below a set amount.