I agree with all you said, with (possibly) one exception. You wrote you were "pretty sure a re-install of Chrome would do the trick". It may not necessarily be the case. There may have been system modifications made. (Your Hosts file, ONLY as an example) I hadn't realized until today, just how much of a problem Babylon can be.
Some questions, thoughts and suggestions.
To rule out any other malware, I would first suggest scanning with the following.
Malwarebytes' Anti-Malware (Free version)
ESET's Online Scanner - Their FAQ and Help sections should answer any questions you might have. (Temporarily disable your A/V prior to running the scan)
You wrote, "In my case my home page appears to be set to Gmail, as it always has been, but in fact what happens is that at start it goes straight to Babylon search".
Would you be so kind as to clarify what you mean when you say "at start it goes to Babylon". When you open Chrome, do you see Gmail at all? Or are you presented with the search page?
You say there is no sign of Babylon listed anywhere. I take this to mean your search engine is listed "as it should be". Have you checked the settings (and extensions) in your other browsers? Is Babylon listed as one of the Search Providers (when selecting Tools>Manage Add-Ons) using Internet Explorer? When you originally went to Programs to uninstall it, could it have been listed as "MyBabylon"?
Check Chrome's LAN Settings to make sure it hasn't been changed.
There are some tools, which might be able to spot the problem, without having to jump through too many hoops. Unfortunately, we don't utilize them here. If it comes down to it, I can direct you to some forums which do. To make sure this a "lone culprit", give the above scanners a try.
I wish I had that "easy fix" you were looking for, but I don't.