You're getting a bit of pushback on this sentiment, so I'll play devil's advocate a bit here.
I've tried gradual refactors in the past, with poor results, because unfocused technical teams and employee turnover can really kill velocity on long-term goals that take gradual but detailed work.
That is, replacing all those v1 API calls with the v2 API calls over five months seems fine, but there's risk that it actually takes several years after unexpected bugs and/or "urgent" feature releases come into play. And by that time, you might have employee turnover costs, retraining costs, etc.
I'm just saying the risk equation isn't as cut and dry as it seems. There's is survivor bias in play in both the "rewrite it" and the "gradually migrate it" camps.