If I may add something, too, I'd suggest that some of the ways ordinals are presented tend to make them sound more 'suspicious' than they actually are. At least, I think there are other ways of presenting them that make them seem less suspicious.
For instance, I think it's pretty common for people to say or imply that using transfinite ordinals involves doing some steps in a process 'more than' infinitely many times. Like, if we have universes indexed by ordinals, we might talk about the step of constructing a universe containing some sets, and if one of those sets was itself a universe, the 'new' one would be indexed by the successor of the 'old' one. And for indexes of ω and above people might say that we have 'iterated the universe construction' more than finitely many times. That sounds wrong from a perspective where you can only "do" finitely many things.
Instead, I think it's good to think of limit ordinals as being more flexible than e.g. natural numbers. They allow you to describe inductive processes where you are able to make local choices about how big your induction needed to be, whereas to do the same with naturals, you essentially have to figure out all the choices that will be made ahead of time, and construct a corresponding natual number that is big enough for all those choices.
Or, in the universe example, one way in which the universe $U_ω$ is flexible is that for any $U_n$ below it, there are other universes $U_{n+k+1}$ above it, but still below $U_ω$. This is not (necessarily) because we have "done" infinitely many universe constructions, though, but because we have defined $U_ω$ in a way that we can talk about however many finite lower universes as we need, when we need to. We will only ever require finitely many in practice.
Perhaps for any particular example we wish to examine, we could construct exactly the finite structure we want ahead of time (like, you must step from $U_{n+5}$ to $U_n$ because I need that much extra room). But it is much easier to construct a flexible structure that is still used in a finite way.