@SilverSpooned My only experience is with a post-paid plan (T-Mobile). The transfer from current phone (XS with physical sim) to the new phone (16 Pro, e-SIM only) was very simple, and the e-sim in the new phone was activated automatically. To repeat, correct, there is no physical SIM in the new 16 series phones, in the US anyway. The caveat here, what I'm talking about, is this was on a post-paid plan with T-Mobile transferred to new phone. I don't know how this will work with your carrier and the type of payment plan you are on. I'm sure there is a solution, I just don't know what it is in your case. (If I had to guess, you would probably add a balance to your account online, after setting up a payment method, etc. Again, just guessing.)
If I were in your shoes I would talk to your service provider. Alternatively, your provider probably has the information online, but my sense you probably need to talk to a customer rep with your provider -- unless someone else her can give you specific help.