score:4
In principle, you are indeed allowed to spend another 90 days in the Schengen area, because the time spent under a long-stay visa or residence permit does not count towards the maximum duration of a visa-free short-stay, see Does tourist visa (90 days) apply after a long-term visa ends in Schengen countries?
One issue will however be how those 90 days are counted. When crossing the border, the guards might ask about missing stamps (also on the way out!). You can then show your expired residence permit to prove that you were legally present in Finland even though your last entry stamp is older than 90 days. But if you don't have any document to account for the time between the expiration of the permit and the moment you cross the border, then that time will probably count towards the 90 days. And if the permit's expiration date lies more than 90 days in the past and you spent the whole time in the Schengen area, then you have a problem.
If at all possible, it could be easier to leave and enter the Schengen area through Finland, as the Finnish border guards might be more familiar with your situation or perhaps even be able to verify that you have an application pending. Traveling to France or Germany from there shouldn't be a problem after that.
Upvote:4
Answering my own question to say what happened.
The police give you a document (with a Finnish name I have forgotten) saying that your renewal is in process, and you can show it when entering the Schengen area. As it turned out, the Immigration Service processed my renewal much more quickly than estimated, so I didn't have to test this personally. However, a colleague at work who wasn't so lucky tells me they had no problem using the expired permit and Migri's temporary document.