Drafts uses iCloud, which is able to sync in the background and download files on demand. Obsidian Sync only gets to sync when Obsidian is open. It might be worth trying iCloud if you haven’t already. (I’m not totally sure it would be faster for you, because it likes to offload files from the device, which Obsidian then has to ask it to download when the app opens.)
To quickly see Sync’s status on mobile you can use the snippet from this feature request to make it always visible: [Mobile] Make Sync icon always visible