As of iOS 10.3 this is mostly fixed:
Hi samuel,
This is a courtesy email regarding Bug ID# 27166979.
Engineering has provided the following feedback regarding this issue:
Engineering has the following feedback for you:
Thank you for you feedback. This was fixed in iOS 10.3.
Latest version:
iOS 10.3.2 GM (14F89 | 14F90)
https://developer.apple.com/download/
Posted Date: May 15th, 2017
We are now closing this bug report.
If you have questions or comments about the resolution, please update your bug report with that information so we can respond.
I confirmed that scroll is not going all over-the-place when you start entering text. (repro is to use iOS 10.3 in “desktop” mode, scroll down page and click on search text input)
position fixed is still somewhat finicky, but the killer bug we had appears to be gone.