Your default Cave is local by design
The default Cave setting is specifically designed to give each browser or device its own CoW startup page. Fleet syncs your collection, but the Cave that opens first is saved separately on every browser and device.
Start in the Cave that fits where you are
For example, you can open a work Cave when you start CoW in your office browser, a personal Cave on your home computer, and a travel Cave on a laptop. Your Caves and bookmarks stay in the same Fleet collection; only the starting Cave changes.