Nope, apparently item_teamflag has higher priority, and it's not just the HUD, but also the gamemode logic (preventing respawns, etc). I once experimented with a version of arena_sawmill that was supposed to support bots using that method, only it didn't work for the listed reasons. But I'm not an expert on setting up logic for custom gamemodes, so maybe I did something wrong.
TF2 is probably looking for map variations based on the 4 major gamemodes, which are:
1. CTF
2. CP
3. Payload
4. Arena
Which means it probably sees an item_teamflag and says "Oh, this is a CTF map"
Since the map controls the goals in the first 3 map types, you just need the HUD override to change the map type.
Also, as weird as it sounds, all game modes except Training are classified as one of the 4 basic types above internally. MvM is a CTF variant, KOTH is a CP variant, etc... The game classifies them more specifically from there. (Player Destruction may be an exception, I'd have to use a server plugin to check if they used game mode 5 for it)
Control Points is even weirder as only KOTH has an entity to tell the game which type you're playing. To the game A/D CP, 5CP, TC, and Domination are all the same game mode. And yes, this means you can make TC with any symmetric game mode and not just 5CP rules; see
tc_meridian to see how it works with item_teamflags. Although, do note that it might take a bit of work to get it working with PLR.