Outlook on the web -- Search tools appear whether I want them or not!

The Search Ribbon tab is indeed rolling out to Outlook on the Web for Business/Enterprise/Education. As far as I know, it is not yet rolling out to (a large part of) Outlook.com accounts.

This experience makes it more in line with how it works in Classic Outlook and now New Outlook as well.

At the moment, it is indeed not configurable, and you're taken to the Search tab whether you like it or not.

Note however that any keyboard shortcuts still work and that the right click menu on an email is pretty much the same as the Home tab for all major actions. This might be an alternative to you instead of switching to the Home tab. To see available keyboard shortcuts, press the question mark key on your keyboard. 

You can provide feedback about this feature change and suggest improvements by using the Feedback button on the Help tab.