Partial workaround:
(works but is not sufficient to fulfill scope of this request)
Position of caret is preserved when we switch between files open side-by-side, by hotkeys.
I utilize Single key hotkeys with Cycle through panes (plugin) in Sliding panes / Andy mode plugin which makes it usable to work with multiple panes open side-by-side.
The following allows to preserve caret position when switching by mouse instead of hotkeys and/or between distant panes (works from https://forum.obsidian.md/t/obsidian-release-v0-10-13/12492):
Pressing Tab after typing in the file name bar should also focus on the editor.