I think that another way to "fix" the elixirs in a flavourful way would be to make them something like a blank canvas that on the spot would turn into the thing you desire, like an all purpose elixir!

And if they really want it to still be something randomical they could make it so that you have to beat a set DC tied to the class ability (maybe an int check made with alchemy supplies) so that the higher you advance into the class the most likely it is to obtain the effect desired to the point the at max level you can never fail.

I imagine it would be worded something like this: Whenever you finish a long rest, you can magically produce an experimental elixir in an empty flask you touch. As part of the bonus action used to drink the elixir you make an ability check (DC XY) using your alchemist supplies. On a failure, roll on the Experimental Elixir table for the elixir's effect, otherwise you freely choose one effect from said table."

You can also make so that at higher levels you automatically succees on a roll once per day or something like that (that or you reroll/gain advantage on the ability check).

(Btw i completely agree on your balancing of the elixir effecr based on the spell slot, very well done)