I too want a WebDAV sync option that will allow me to sync an Obsidian vault held on one machine to other machines and to mobile devices. I can’t post in the Mobile section but as an example of a system that works is how DEVONThink handles their sync using WebDAV. The DT databases and files are on a local machine. There is a Sync store created on the WebDAV server that other devices can point to and update. You can select which of the various DT databases you want to go on which devices, convenient for putting only some on a mobile device and perhaps all on say a desktop and a laptop.
I’d love an equivalent option for Obsidian and Obsidian Mobile.
PS another application that allows for WebDAV sync is Omnifocus. Same basic thing you set up a link to the server and your data are available there for other devices to access.