I want to be able to use that particular phone number with Twilio, so it needs to leave Verizon. And, yes, I really want that result.

Then to keep that number, you will have to port it out.  If under contract, you will have an ETF.  If not under contract, then you can leave whenever.  So, after you port out, you will have your number on Twilio and then you can get a new line and phone with Verizon, or get a second line before you port out to make that easier.  However, doing so will either get you into an Edge payment, 2 year contract, or BYOD to a plan or prepaid.

Right, that's why I was asking about a way to do this without an ETF. I have two other phones and a tablet on Verizon, it's not that I'm not going to be paying them into the indefinite future, but I just need this particular phone number to be somewhere else and don't really want to pay $250 to make that happen.

Unfortunately, that number is tied to that contract or line of service.  Even if you changed phone numbers and your number went into a pool again, it would be a while before it was released and doubtful your could pick your own number.  Verizon has that trick figured out.

It's not like they really care what number I use, because they'll cheerfully assign me a completely different phone number to use with that contract obligation. And it's not like they really care which phone instrument I use, because they'll even let me swap phones online through a web interface. So all they've really got (or appear to care about) is that I promise to maintain service with them, using some phone number, and some phone, for the next year. And I've got no problem with that.

I'm not trying to avoid paying them, I'm just trying to avoid paying a termination fee for a contract that I don't want/need to terminate.

I understand you have no problem paying Verizon, but options are limited when you want to keep a particular number, but want to use it elsewhere when it is under contract.