Yes, Bluetooth introduces a tiny bit of audio latency, which is perfectly acceptable for 98% of all use, and 99.9% of RasPI 400 uses. For the rest of the cases, use one of the other RasPI400 options, a RasPi 4 or a Razer laptop.
I have an AppleTV4K, and everything is perfectly synced with at least 3 brands of Bluetooth headphones - an 3-year-old eKids one, a 1-year old TaoTronics one, and a 2-year old nameless one.
In the early PlayStation 2 days, many flatscreen TVs (both Plasma and LCD) had delays that made games like Guitar Hero unplayable, so the games added a latency estimation/calibration setting you could use; My 12 year old Onkyo HDMI receiver has a special "eliminate latency in game mode" setting which disables some processing features and reduces sound latency by 50ms or so.
It's not that there were never problems; But anything made in the last 5 years or so, especially using BT4/BT5, should not have these issues.
Edit: just tried with an 8 year old or so JBL Clip I found lying around, and it does have ~100ms latency (estimated, not measured). But the newer ones don’t have it.
I use mpv and whenever I get audio latency from my headphones* I just adjust in the player.
* - It seems like there's some adaptive algorithm where they start off at 0ms latency but if they experience one hiccup they increase to 250ms-300ms and then another hiccup causes them to jump to 500ms. This is also why a keyboard shortcut that let's you adjust the exact a-v delay value is useful beyond just setting the value once.
