Those services support Linux, but like you noted, not all the games that use the services do. (adding to your Destiny example, Fortnite has explicitly been denied[3] Linux support.)

[1]https://nitter.unixfox.eu/thebattleye/status/144147781631129...

[2]https://areweanticheatyet.com/

[3]https://nitter.unixfox.eu/TimSweeneyEpic/status/149051919491...