I think that might be a bit too much of an overcorrection, so I might think about handing out 1/2 proficiency bonuses at 6th and 18th and a full proficiency bonus at 12th. (or probably more likely - raise the first 1/2 proficiency bonus to full at 12th and give a second 1/2 proficiency bonus).You might hand out proficiency in a new save at 6th, 12th and 18th levels. By end game you'll be proficient in 5 out of 6 saves that way.
But what I actually do in practice outside of when I'm teaching kids how to play (where I try to go as by the book as possible) is try to have two different possible saves for effects depending on the effect. Especially at higher levels. And allow players to propose ways they might try to save against a particular effect if they have an idea I didn't think of (you have to be able to say no a lot at first depending on the table - you can't use Int for all of your saves just because it's the best one you have no matter how much you push it). With the right group it seems to help though it doesn't completely eliminate the issue at high levels.