Agreed. Even though on the jets I’ve flown, they’ve varied in the before takeoff checks from “APU-OFF” to “APU- As required” (or similar verbiage), I leave it running until 10k, securing it while doing the other 10k checks such as pressurization check, etc. There’s no penalty to do so, and many potential benefits in doing so, as you’ve mentioned.
I think the problem comes, when people see “APU-Off” before takeoff, that they read it as a must-do, rather than an option or reminder. Especially with some of the pilots I’ve flown with on one particular airframe, who are hardcore about not using the APU in the air. That’s a story in itself.