> increased home values, allowing them to sell and improve their lives
That also raises property taxes, making the neighborhood unaffordable and driving them out.
> it's now a more pleasant area to live in.
For new wealthy residents. People who have spent lifetimes there don't want everything to change and have their communities destroyed.
> Yes, they may have to move, but also the places they move to on their current budget may be nicer - because the people who can afford better have moved too.
These are theoretical and very general averages. The actual individuals often do not benefit. Being forced to move is not a mere inconvenience to your theory.
The alternative: new housing doesn't get built. Existing housing - including the "bad" neighborhood that isn't redeveloped for fear of "gentrification" - gets bid up to the moon. People who can't afford rent end up moving anyway and commuting from very far away, if they're lucky. Or they end up on the streets, if they aren't so lucky.
That isn't theoretical. I just described the SF Bay Area.