You make the claim (in the form of a question) that arbitrarily assigning frequencies to note names in the chromatic scale would "solve the problem of just intonation sounding different in every key except 1 and equal temperament being slightly out of tune in every key".
Based on this statement is seems that you do not know how these tuning systems arise.
Just tuning is based on the natural harmonics of some typical vibrating systems. Hence intervals are very "harmonious" in this tuning system.
The harmonic sequence is fn = n*f1.
From this we can get the "5th" and the "3rd" from n = 3 and n = 5 harmonics. Obviously this is not the correct ratio but if we bump them down into the first octave [f1, 2*f1] we get f(5th) = 3/2 * f1 and f(3rd) = 5/4 * f1.
If you apply the same reasoning starting from the 5th you get the ratios for the 7th and 9th (or 2nd bumped down). The "4th" is really a 5th below the tonic so we require the ratio of 4th (octave lower) to the 1st to also be 3/2, which becomes 2/3 upon inversion, and 4/3 when moved up and octave. The point being is that these ratios are based on the physics of vibration. This produces a set of notes that have THREE distinct consecutive ratios, the half step = 16/15, and two types of whole tone with ratio 9/8 and 10/9. For example the ratio Re/Do = 9/8 but that of Mi/Re = 10/9.
In terms of letter names perhaps we had chosen too few in the early days of music, or perhaps we had some other notation not currently in use that helped us distinguish these. If one wanted to build a D scale using, as the starting point the second note of the C scale then the second note, Re, could not possibly be the E of the C scale because it would not have the correct ratio. This is sometimes "corrected" by lowering the second note, and likewise for the others that do not follow a strict pattern. This "correction" helps standardize things and allows us to use a very simple alphabet for describing the notes available to us.
So, when you say that the Just scale is "different in every key" it is not clear what you mean! If the ratios are kept true then it should sound THE SAME in every key. I think you need to be clear about what quality you think is different.
The 12TET system defines the half step as the 12th root of 2, ~1.05946309436... . This is an irrational number and hence impossible to calculate exactly, though we try our best. In this tuning system ALL consecutive 1/2 steps have identical ratio. Hence ALL whole steps have identical ratio regardless of where you start, r ~ 1.0594631^2 ~ 1.122462. By the way 9/8 = 1.125, and 10/9 ~ 1.111. All one needs to do is get 1/2 steps to register the same value to within the precision of some spectral analyzer. Then everything is "in tune". In theory one could tune in 12TET with enough precision that a human could not detect the drift all the way through the spectrum of human hearing, to within the pitch discrimination ability of the human ear and brain. This is not possible, imo, out to infinity but it is possible for a finite bandwidth. So again, what exactly is "out of tune" for the equal tempered scale? Is "out of tune" your way of saying that the tones are not based on harmonics of the fundamental, dominant and sub-dominant?
I think that you need to enhance the question to be more clear. However, based on the two mathematical definitions of tones it is simply NOT possible to (1) make the steps equal ratio in all places while maintaining the harmoniousness that naturally occurs when harmonics are used. I am not sure if this helps answer your question but I've tried to interpret it faithfully.