The main use case for biometrics at the border is matching you to the documents you present (biometric passport, visa, residence permit), i.e. one-to-one verification. This makes tampering with the documents or using someone else's documents much more difficult.
Many French residence permits do include a digital photograph and at least two fingerprints and use the EU common format defined in regulation 380/2008 so it is conceivable that border guards in Amsterdam or Budapest could read them and take your fingerprints for comparison but this is not systematic and there is no obvious link between this and the renewal status of your permit.
Starting in 2020, the EU's Visa Information System (VIS) was also updated to support long-stay visas and residence permits so it's also possible for member states to share these data with other Schengen countries and provide another way for border guards to verify the authenticity of your permit. For residence permits, unlike short-stay visas, this is however optional and I am not sure France is currently using the VIS for this purpose.
By contrast, one-to-many biometrics search (identification) is less reliable and hard to fully automate (NIST publishes extensive tests on commercially available facial recognition algorithms, research on fingerprints is less active but I believe performance is even worse). I found no evidence the French AGDREF database (which contains both the reference copy of the biometrics enrolled when you applied for the permit and information about its validity) is used for this purpose.
For all these reasons, I would be very surprised if border guards anywhere would routinely look up people by their fingerprint or photograph and I see no way the renewal issue would come up through a biometrics check. Besides, the end of validity date is printed on the card and would be obvious to border guards anywhere, even without any access to the biometrics recorded on the card or in official databases.
Importantly, it is not illegal to travel or apply for renewal shortly before the end of your permit's validity. The reminders come earlier because they know it takes time to make an appointment and get the permit and they have rather have people carry an actual permit instead of a temporary document (récépissé) but it's only if you apply for renewal after your permit actually expired that you would have to pay a €180 penalty fee.
Until then the permit is valid and sufficient to exempt you from any visa requirement. The only thing that needs to be valid beyond the date of entry in the Schengen area is your passport and even that requirement ought to be waved if you are on your way back to your country of residence.