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.