The original Phantasy Star for Sega Master System and Revelations: Persona for PlayStation both have one horrible, horrible flaw for me.
The battle music.
Revelations: Persona's music isn't abhorrent as it is insipid, and there are a few good parts in it. It's just, in a normal battle, you're not likely to ever hear it, since everything will be dead within 20 to 30 seconds. It repeats the same few riffs repeatedly, and it was maddening after a time. It got to the point where I would literally set the team on auto-battle and leave the room. As much as people decry its PSP port, Shin Megami Tensei: Persona, for having music in the vein of the later two games, I will always stand by the fact that "A Lone Prayer" is better than "Battle".
For Phantasy Star, I mostly think it was the arrangement. Its remake, Phantasy Star generation:1, has pretty much the exact same song, just with a different, more modern soundfont. I found the other songs and many aspects of PSI great, it was just this one, unfortunately often-heard, track.
Other than those isolated instances, I can't really think of a time where I couldn't look past tiny flaws in video games. And both times, it was solved by more up-to-date versions.