You need to look into the cost of making the floor sound. If there are only a few weak spots, maybe they can be 'shored up' from below or you could have the strips/planks removed from above, and go from there. Depending on how pronounced your 'weak points' are, it may be best to simply remove what you have and install a 'sound' sub-floor where you can 1) consider reusing the existing hardwood, or 2) going with something new all together.
Since they are stained and splattered w/paint, in the least, a refinish is in order. BUT, I would like to warn you that at the 'weak points', the sanding may NOT be very good, thus, creating a sanding/finishing problem(s). If I were you, but I'm not, but if I were, I'd look into seeing if the existing hardwood can be saved (IF it has some 'inherant value), and 2) adding a sound subfloor. My best to ya and hope this helps.
Jay J -Moderator.
PS: IF you decide to rip up the existing floor and save it for re-use, I wouldn't finish the floor until you have the hardwood re-installed ...