From the looks of the gate, you'd have to rebuild the gate to get a decent hold on any latch you reattach.
The wood looks rotten as well as already having multiple holes in it from previous attempts at installing latches.
The right-hand gate was made improperly to begin with, so there's extra stresses on it and the latch that can be reduced by reversing the direction of the diagonal brace. You can attempt to address that with adding another brace or a cable in the opposite direction, but that still doesn't fix the problem of the disintegrating wood.
You state that you're trying to avoid replacing the gate and just want a band-aide solution. Unfortunately, that's no longer a practical solution. If you don't replace it now, you'll spend more time f--, um, fiddling around trying to get the latch to work than just replacing the whole thing would take.
Looking at it, you might be able to reuse the decorative verticals on the outside of the gate and just rebuild the frame, which is as simple as making the same rectangle with some new lumber and adding the brace from bottom left to top right, instead of the current top left to bottom right. The current brace wants to kick out the top left because of the weight below it. But switching the bracing will tend to suck in the bottom left and support the weight above it. Yes, this means repositioning the spring, but it'll make for a much stronger and less maintenance gate.
Using through bolts is one way to make this latch better, just make sure you use washers or carriage bolts to help prevent excess damage to the wood when tightening them down. Also, you might want to box in that top left corner to give yourself more area to install the latch. That seems the likely reason the brace was installed wrong, but that's a bad reason to install the brace wrong. You can simply add more lumber there and/or add some flat steel (or aluminum) sections to increase the area the force of the latch acts on.
Bolts and screws aren't really made for lateral/shear forces, just clamping forces. Once you have lateral/sideways movement in the fastener, it's going to break. It's also going to make the hole wider so the screws and bolts just fall out, which is what you are seeing. By adding the metal and/or extra lumber, with washers on the heads and nuts of the bolts, you can tighten down the bolts considerably and prevent crushing the wood as well as lessening the shear stresses.
You'll also want to use some method to lock those bolts into place. Thread locking liquids are good for that. Using the medium grade (blue Loctite) is more than enough for this purpose. If you use anything more than that (red Loctite), you'd better hope you got it right the first time, because the gate will rot apart again and the bolts will rust to dust before the thread locker loses grip.
Or you can use locking nuts or washers. IME, lock washers are hit and miss, but mostly miss. Wrenching down too hard on them can eliminate their usefulness completely, or simply just destroy them. A nylon insert locking nut (sometimes called a nyloc), is a better option than a lock washer. The problem with nylocs is that they have a limit to how many times they can be installed/removed and still function properly. Installing it and removing it once will greatly reduce its locking ability. It'll still work as a nut, but the nylon will have been damaged. Installing and removing it twice will essentially make it not a locking nut anymore.
Once you have the right-hand gate replaced, you can use the same techniques to rebuild the other side. However, that gate was built with the brace in the proper direction, so use it as a template to do a straight forward rebuild with the same geometry. The only thing I'd suggest doing differently is to not leave the one hole in the latch hanging out in space. Use some lumber and maybe some of the same metal bar stock to support that corner, put another fastener there, and reinforce the whole latch.
Using reinforcements on both sides of the latch, as well as getting the diagonals in the correct alignment, will keep that gate straight and working correctly for years to come.