I think this change would be fine
A common complaint about the Assassin roguish archetype is that its level 9 and 13 features are quite lackluster. They only make sense in specific scenarios, which may not come up in your campaign, making them completely useless. I agree with your player that Supreme Sneak does suit the Assassin from a flavour perspective, and if it helps that player to feel like their choice at level 3 wasn’t a mistake, I don’t see a problem with it.
Mechanically, it is a straight up trade. They aren’t gaining Supreme Sneak in addition to Infiltrator Expertise, it’s instead of. It also doesn’t have any multiclassing issues, such as giving a feature from a different class to this character; any character would need 9 levels in rogue either way. This isn’t a particularly strong feature, like the features that roguish archetypes gain at level 3; their level 9 feature is usually somewhat of a ribbon feature. Rogues generally are also considered to have a strong “core class”, so they don’t gain that much additional power from their archetype features.
I also don’t see there being a balance issue in combining Assassinate with Supreme Sneak. Although I agree with your player that Supreme Sneak suits an assassin, I don’t think it leads to any game-breaking combinations, since an Assassin rogue is very likely already extremely sneaky (proficiency and likely Expertise in Stealth; credit to Cubic for pointing this out first, but I wanted to back up my answer with this point also).
The only thing I would add is that it might be worth making sure your other players are alright with this change, especially since I imagine your Assassin player is the only one who will be getting this change. It shouldn’t become a free-for-all where people can basically just mix-and-match their favourite features from all of their class’s archetypes. Assassin is somewhat of a special case because of how weak their level 9 and 13 features are considered to be, given that they are very campaign-specific.