I know how you feel, sometimes it's just incredibly disappointing to lose a game just because of bad luck. I don't know if one "pick any card you want" mulligan would imbalance the game. Especially in mirrors it's very often luck if both players are equally good. It's just the nature of a card game I guess....So, another game, another RNG loss. Mirror SK Greatswords (but the guy used Patricide for some reason). It all came down to a Greatsword behind a defender. He drew his Gremist and Decree, I didn't. He could Morkvarg my Greatsword, I couldn't his. Could not have been won. And the guy made a few misplays like Decreeing a unit he didn't need. Plus he used a leader with less synergy. No advantage has any meaning if you don't draw what you need.
Another layer of RNG is how important for many decks it is to win R1 (and not commit too much) where you only got about 50% chance to draw the card you need. Losing last say in R3 where you have no stategem or any other advantage except one more mulligan is disastrous for too many decks. And losing round control for decks that got no good short R3 is also disastrous.
So, after some consideration, I agree with @Alexander_Volgin. In order not to break the game implementing some new and untested mechanics, increasing the number of mulligans to 4 on blue and 3 on red is a safe solution that would lessen the impact of RNG.
A fun addition to that would be an option to banish or graveyard one card during each mulligan in order to get rid of tech you know you don't need (or think you don't need and then get surprised - another layer of fun decision making).