You've got several parts to this question, and I'm going to break it into 3. Here are the issues I see:
- Is rolling stats a problem?
- Are high stats a problem?
- What can we do about it?
Is Rolling Stats a Problem?
First, rolling is presented as the default option in 5e, though they do require the use of point buy/array for organized play. But, when you read the PHB it starts by explaining rolling stats and presents point buy/array as a secondary option.
However, the big problem with rolling stats is not the situation you're describing. The problem is a different one. It's when one or some but not all characters have ridiculously high stat and the others do not. Or when that one guy rolls 2 8's a 9 and three 12s.
When all of the PCs have similar stat arrays, which is the situation you're describing, there's not a problem with rolling stats.
Are High Stats a Problem?
In a game where all PCs have rolled their stats well, and you have players starting with 18 or 20 in their primary stats almost universally, there are some potential problems.
The biggest problem I see is that you've removed one of the major choices characters make "do I take the stat increases or the feats." However, while this is an important choice, it's not one that is going to break your game if you remove it.
As far as the math goes, your characters are both more survivable and better at their jobs at L1 than they would be with an array/point buy scenario, but it's in measures of 5-15% which although it's meaningful, is not something that is going to break your game.
My recommendation with this is that if you don't want to reroll or adopt a point buy/array situation is talking to your DM about playing the first few encounters straight and then increasing the difficulty more rapidly. If your PCs are hitting to easily, it might be proper to more regularly give enemies better armor or more hit points.
What can we do about it?
This is something you need to communicate with the group. Do you collectively feel it's a problem? What kind of game are you interested in playing? Does the prospect of being better than a normal L1 excite the group in a way that point buy/array just simply doesn't?
If the group is feeling the same way you are (that you're all too powerful and need the nerf bat), then they would probably want to reroll/switch systems to something more equitable that produces more predictable results. However, there's a strong chance based on what you said that they want to keep these.
I'd advise against using an RP nerf to compensate for good stats. It would be much better to simply decide to reroll your stats or use a point buy for your character. However, ultimately, the problem is a group level problem and you choosing to do something about your specific character is only going to cause problems in the group (either your effectiveness will be diminished or your RP element may be something the characters find annoying). At the very least, run this by your DM/group and make sure it's something they buy into.
Overall, what you do about this largely depends on your place in the group. If your the DM it's well within your rights to punch the brakes and say "hey, these are really high, let's make sure this is what we really want." But as a player, you can do the same thing, but if you get pushback, you probably need to understand why your group is happy with this and you are not.