I'm not a boat person but it feels like you've got conflicting ideas about whats possible. You said:

Because of-fucking-course anybody who wants to buy and live aboard a cheap (easily $50k or less) old sailboat instead of paying rent forever

...and...

the ideal goal would be to be able to do things like work remotely and keep my kids entertained while circumnavigating

I don't think you're going to find a $50k boat you can buy (and maintain!) that can house four people comfortably for transoceanic cruises while also affording you the ability to work a full time job from the boat. I would think you're looking at a MUCH larger boat, possibly with some full time crew to accomplish that, and at that point $1k a month for global high speed low latency internet is probably a a small fraction of your monthly expenses.

Look into a low-power server for your boat, try to host as much of you can locally. I'm pretty sure you could fill mot entertainment needs that way, and 'top up' your content via terrestrial Internet when you resupply.

Then it kinda depends on what you need for work. Upload/download code snippets? Video conferencing all day every day? There's like a big span for the bandwidth you might need.

work remotely and keep my kids entertained while circumnavigating

FWIW depending on your work you can do a lot of that on the cheap, namely if you work is not heavy bandwidth or latency dependent, code and voice do not take much. You can get a lot of resources offline too, e.g. Wikipedia, Stackoverflow, etc in a convenient package with Kiwix. Download this at the port or prior to the legs of the trip where you don't expect to have good connectivity then update at the next point. It's honestly a matter of hours at most. I do it before every trip and it gets easier every time.

My suggestion anyway for kids entertainment is also offline entertainment, e.g. GCompris but even content. Again you can put Wikipedia from Kiwix on your then local WiFi (no Internet, just all devices on the boat) with a small RPi Zero (low energy consumption) with a 1TB microSD card (so cheap now!) but also a media server with all the videos you want from Internet Archive. There is a TON of content. Once there they can watch with any media player that supports network play, e.g VLC or mplayer.

TL;DR: 1TB from the middle of nowhere on the cheap is indeed tricky but 1TB from a good connection THEN go offline is actually both very easy and more than enough to be entertained for months, if not decades with e.g. Gutenberg project!

I'm just going with an Iridium for calls and short texts. I save up all the bigger missives for when I hit wifi. That doesn't work for most work situations I think.