Hand sewing mostly likely will result in the stirrup leather breaking while you are riding. So to play it safe, I recommend getting them properly fixed with a leather sewing machine or buying new ones.
I resewed several bits of tack "back in the day" and it was at least as strong as the original. I had old stuff where the thread had rotted away with time. If you use the proper thread (heavy waxed thread) there's just no way for the thread to give again, at least in the next few decades (like the original), and I always reused the old holes so there was no further weakening of the leather. So no reason it would be more likely to break than before.
Also, there's absolutely no reason for hand sewing to be weaker than machine sewing - in fact less, as the 2 threads are independently weaving back & forth through the stitch holes, not depending on an interlock of the 2.
I did find it to be a fair amount of physical work, pulling the needles through with the doubled-over thread where it passes through the eye - pliers were necessary. Just to warn you! But then, I usually didn't really open up the old holes any farther.
One thing I didn't know at first (until I read about it in an article - this was the '70s, no internet!), is that when you've pulled one end of the thread through the hole, and are about the pull the other end through from the other side, be sure to pull the first end back toward the direction it came from - so it's like a you - then you can pass the needle with the 2nd thread end through the hole without passing it through the thread already in the hole & weakening it. Hope that made sense. Actually, this guy passes the 2 needles in the hole so that way the needle can't spear the thread because the thread's not there yet:
He's so fast I couldn't even see what he was doing until several stitches in! Of course, it's probably way easier with soft shoe leather than with thick tack. Plus, he's really really good!!!
So good luck, Moonshadow!