I think a distinction needs to be made between aged and dating. TOS has aged and it looks like a series filmed in the 60s. But I don't think it looks dated. Then again I grew up in the 60s and 70s.
However, I feel the first 2 seasons of TNG look dated but the rest of the series still looks fine.
I can look past cheap sets and effect and things that were products of their times like fashions and hair styles. Most of my problems with TOS come from the acting and writing. Poor acting or over acting combined with poor dialogue could often make things unintentionally silly. I usually find Shatner to be the most off putting especially when he's trying to be the big leading man of the series. I always found the difference between Kirk in TOS and the movies to be pretty striking, either Shatner became a much better actor or he decided to play the character with more maturity. Other things that bothered me were clumsy looking action scenes and overly dramatic music and sound cues. I'd actually like to see modern editors and sound mixers take a pass at TOS because I think most of these are things that could be fixed.
The first 2 seasons of TNG were pretty dated even back when they were new because the writing was poor. Sure the costumes, lighting and direction all got better but the biggest change was the writing. Season 1 and 2 were trying too hard to be TOS which just didn't work with its large ensemble cast of undeveloped characters.