Judging from the interface, one should get a reliable number for all publicly viewable manuscripts in the database through the "advance search" page, which is accessible from the starting page.
There one can select the option "public view". Searching just with this option set to "yes" returns 17290 results.
There's a second relevant yes/no search option, "digitization". A search with both "public view" and "digitization" set to "yes" returns 17258 results, one with "digitization" set to yes and "public view" set to "no" returns 19396.
An "advance search" without any values at all entered returns 5145280 entries, by the way.
Perhaps registered accounts get to see more? However, one needs a "permission letter" to register ...
Best regards,
Birgit Kellner