31.5" was the standover height of the frame of my vintage Raleigh Team USA, and I could only straddle that with cycling shoes on. I am also 5'10". The top of my saddle was at least 6" higher than that. A 31.5" saddle height is almost physically impossible with any properly sized frame for the o.p. Absolutely slammed, I would imagine a minimum possible saddle height of 35" on my frame which was admittedly too large, but assuming a proper 22" frame (vs 24") well you can see how things scale down. I usually shoot for a center of BB to top of saddle measurement (tape measure angled to follow slanted centerline of seat tube) of 28.5" on all bikes regardless of crank length, then I adjust for comfort (+/- 0.5"). Obviously that would vary by person, but NOT 6" worth of variance! I always send people back to measure again (and again, and again) when their result are as outlier from the norms as in the present case.
Measuring cycling inseam ideally requires two people, but it can be done by oneself. In a pinch, add 2" to your tailors inseam (if male) and you wind up in the ballpark of a cycling inseam. The best way to measure cycling inseam is with a thin, but rigid, book that can be slid up alongside the ... package, and nestles in the crotch, right up against the bony structures. In bare feet, measure from the floor to the top of the book. Do that twice more. Average the results. There you are. Informed with a fairly significant, but useless, piece of information. Here is how you really find your ideal saddle height: center the saddle on the seat rail. You should have about as much rail ahead of the clamp as behind. Tilt the nose of the saddle up slightly. If you have a saddle where the tail is kicked up, you may have to tilt the nose up even higher. You are aiming to have the area of the saddle you sit on, dead level.
Now straddle the top tube in footwear. The nose of the saddle should hit you right about at the top of your ... split. Now, get your usual take off foot up to 10 or 11 o'clock on the crank and push off. As you are rising (you should be rising) to the saddle, note that you should find the saddle 'right there' when your trailing foot comes up and you are in the saddle. It shouldn't have required any 'skootching', up or down, to find the seat. You should simply be able to start off from a position astride the top tube and land in the saddle without drama. After you can do that you can use any kind of fancy takeoff you have mastered during your formative years.
A 'normal' shoe size for a 5'10" male is between a 10 and 12. At these shoe sizes it should be fairly impossible to put even one toe on the ground and remain on top of a saddle that is properly adjusted. That's just the way it is. The reach to the bars should definitely put between 45 (racingish) and 55 (commuting) degrees of angle in the riders back. Bolt upright should be impossible. Obviously pictures would really help identify issues in this thread. There are, of course, allowed variables for individual tastes but there are ... conventions. There are limits in the adjustments of seatposts and saddle clamps and also stems and handlebars but if one starts off with a frame in the ballpark, a workable result should be possible.