In my experience, people who think Sneak Attack is overpowered are either very bad at math or very ignorant of the damage capability of bog standard Fighters that don't use any of their class features.
Rogues without Sneak Attack are worse at combat than Valor Bards by a significant margin. If you don't want to bother making checks, you're not going to break anything, because the non-SCAG cantrip/non-multiclass Rogue will still track behind a damage-focused Fighter (and the SCAG cantrip Rogue will track with the damage Fighter) in pretty much every case except for three:
1. The Rogue consistently has Haste and does Ready Action to Sneak Attack on another creature's turn as well as his own (but a GWM/SS Fighter with Haste would still be doing similar damage without having to Ready Action)
2. The Rogue multiclass dipped 5-6 into another class for Extra Attack (especially if he is XBM/SS, which is by far the most abusable build with an EA multiclass, but an XBM/SS Fighter still eclipses an Extra Attack5/RogueX build unless combined with scenario 1 above, where the multiclass EA5/Rogue does finally eventually win out [until 20, where it's dead even again])
3. You've banned the Sharpshooter and Great Weapon Master feats (which I wouldn't advise because otherwise you're just indirectly buffing spellcasters/nerfing martials) in which case Rogues will be slightly better than Fighters at high levels (until level 20, where again Fighter becomes king regardless).
(sidenote: extra Fighter damage bonuses such as Action Surge, Battlemaster, or crit on 19-20/18-20 weren't considered and will tilt this a lil further in the Fighter's favor, and the Fighter already had the edge to begin with if the Rogue didn't go completely out of his way to be competitive)
So, would Rogues be broken if Sneak Attack were always on?
TL;DR: lol no