Overwatch is a objective-based game, and like Valorant is a "hero shooter". Apex and Fortnite are battle royales.

I think the closest I got was The Finals but still class-based, so reminds me more of Team Fortress.

I loved playing 1v1 on Quake 2/3 and UT, also team deathmatch, from the list you commented it feels like each game got one of those aspects but none that makes the genre of UT what it is: knowing where weapons/ammo/armor spawn, map knowledge to navigate around, emergent movement mechanics (rocket jumps, strafe-jumping, etc.).

Interesting to see this genre mostly died out, and remnants of it have been scattered across other genres.