Both your Apple TV and your router remember their “state” between reboots, so while rebooting your router does not fix the problem, it is possible changing something on the router may fix your problem. For example, as your Apple TV boots up, it will request an IP address from the router. If another devices attempts to use that same IP address, your Apple TV can become disconnected. Rebooting the router may not force all devices to get a new IP address, so the conflict (and AppleTV issues) will remain. However, when you reboot the AppleTV, it will ask for a new IP address, and be able to connect successfully. So the workaround you found (reboot the Apple TV) may help short term, but the fix might be to look at the router configuration (or other devices) that might be causing conflicts. Two ways to test this might be:
Can you try an Ethernet cable temporarily to see if that fixes it?
Can you create a “guest” network on your router (with a different password) and have the Apple TV log in to that?
Both techniques will (most likely) put your AppleTV in a different IP address range, reducing the likelihood of conflict. If this fixes it, it is probably the easiest fix.
If it doesn’t fix it, it would be helpful to know the model of your router, and maybe a screenshot of the connected devices and firewall rules from that router to be able to narrow down the problem.