DEVONthink is not stored on the iCloud sync location. Databases are stored locally and iCloud (or other sync method) only holds data used exclusively by DEVONthink to sync.
You are apparently running into the unreliability of Apple’s iCloud for synching. Discussed t length here and experienced by others. Or perhaps you are using different Apple ID’s. Hard to tell from here.
You don’t say what went wrong with Synology WebDav setup. Works fine here, but it is a more complicated setup than other methods. There are threads here with good advice how to setup. I recommend you search.
Also consider using Bonjour which is the quickest and most reliable sync method. Turn “on” Bonjour on your machine and then setup your wife’s machine to connect and sync.
See the DEVONthink Manual for more elaboration on above, and supplementary info at: