I don’t think this is a DAVdroid but a general “issue” of Android nowadays… Maybe you need to dig into the depths of your phones battery optimization features. I myself had to add DAVdroid additionally to the “Unmonitored apps” list. Then the automatic sync is working without problems.
Another thing that often prevents automatic sync is a firewall, anti virus or privacy guard app. Also some network misconfiguration or proxy settings can be a problem for automatic sync.