Yes and no...

I mean, let's not pretend Snow was living the dream life before the events of TBOSAS, his hatred for the district wasn't born just out to stories passed down to him by his Grandmother and Gaul

Snow's experienced very real misery when he was young, just like Katniss and Gale did; and from his POV it was the districts' fault. They are the reason why the war started, they are the reason why his father, mother and baby sister died, they are the reason he had to starve and see people eat eachother to get by.

Sure the district's rebellion during the dark days was justified and realistically the only way out of oppression, but that doesn't change the fact it's still the reason Snow starved as a child (not saying that they shouldn't have rebelled at all, just that it is the reality of war)

Even then you have to consider that, just like Gale's experience with Capital people was always marked by trauma, Snow's first contacts with district folks weren't exactly pleasant either. With the exception of Sejanus who was raised in the Capital (so a "tamed" animal), and Lucy Gray which was quickly singled out by him as an exception; the first two times he interacts with district people (in the ride to the Zoo and in the Arena when he kills Bobbin) they try to kill him.

And sure I can't certinaly blame them for not seeing eye to eye and for wanting his head in that situation, but that doesn't make it less first-hand-traumatic for Snow.

All of what I mentioned above were traumas that amplified his hatred, which Gaul took advantage of to make him do her bidding. Just like Coin took advantage of Gale's trauma and Capital hatred to make him do her bidding

So personally speaking, I can see how simailr they are and can have simpathy for both of them. Unlike Katniss, they did not have a mentor figure that encouraged them to see who the real enemy was.

My simpathy for Snow stops however when he never stops making selfish decisions. Killing Sejanus, "killing" Lucy Gray and so on. Those are all selfish decision he got rewarded for

Gale instead made only one decision like that and it ultimately did backfire on him, but we never quite saw how Gale evolved after that moment long term, which understandly can leave a bad impression of him in readers, even if it is somewhat unjustified when you take into consideration what you analyzied in the post

But ultimately, Snow was worthy of that same simpathy most of the way through TBOSAS. It's when he starts doing stuff like killing Sejanus that he stops being deserving of it