I have a [retail, boxed etc] copy of Hexmaker somewhere... don't recall whether it was easy to use or not - I never used it but my son did, he was about 10 at the time. Seem to recall it being menu driven. Would have been for Hexen II rather than the original Hexen I think.

Duke Nukem 3D Atomic Edition shipped with some pistol asshole bots and a map maker (BUILD in this case)
There's of course several UnrealEngine games that support bots+ shipping a map editor, but like you said, too complex.
The only "simple" map makers i've seen were the Deathmatch Maker series for Quake and Quake 2.

When I first read the topic title, I was going to mention Unreal 🤣
The only extra suggestion I could make is the Battlefield series of games. They support bots and theres a mapmaker for Battlefield 2 at least.
But weather I would recommend it, is a whole other thing as I found the Battlefield Editor very cumbersome.

It is possible to create your own maps there...but it would be weeks if not months work to finally create one.

>It is possible to create your own maps there...but it would be weeks if not months work to finally create one

Yes, it's the same with the Unreal/Half-Life/STALKER/Deus EX/etc level editors - they are very powerful, but they take real time and talent to use well. Whereas in the Timesplitters game, everything was drag and drop, using a simple GUI, it was genuinely fun to make your own maps. The downside was how simplistic there were in comparison to the more complex and varying maps made with other programs, but even so they could still be complex and varied enough to be really fun to play in. You could knock up a simple map in literally a couple of minutes, or spend an hour creating and fine tuning a map, and enjoy every moment of it (all the more so since the map maker took care of all the annoying problems like bot-pathing and texture clipping).

I've found that Cube 2 (and maybe Cube 1, it's not clear from what I read) has a map maker something like I mean, although it's not as intuitive as the Timesplitters ones, but it is more powerful (to judge from the videos I've just watched). Cube 1 and 2 are available free from:

http://sauerbraten.org/

Also there's a first person shooter creation kit available from:

http://www.thegamecreators.com/?id=2001&m=view_product

Which I *think* I'd heard of before, but had slipped my mind. Anyway, I don't know how it compares, but they have a demo so it will be easy to see if it's what I want.

Genesis3D engine also had some dumb bots and an editor (it's pretty much a base engine anyway)

Well I do recommend unreal tournament (the first one) if you want to make maps. Why? Because it is one of the more simple ones to use.
Once you get more familiar with the editor you can whip out a playable level fairly quickly if you know what you want to make.
Adding but support to a level (depending on the level and gametype) can be really simple.
However making a nice and polished level takes (much) more time due to having to do lighting and texturing + aligning.

In case you want to give it a try, this ->SITE<- has a really nice tutorial for new users and has more advanced information as well.