coefficients not returning coefficients in the coefficient ring

[pzinn]

I recently got reminded of an issue that's been bothering me for years:
why does coefficients return the coefficients of a polynomial not as elements of the coefficient ring, but rather of the ring itself? this makes no sense to me. Maybe at some point promote didn't work well and so it was more convenient this way?
Example borrowed from the doc:

i1 : R=QQ[a,b,c,d,e,f]

o1 = R

o1 : PolynomialRing

i2 : S=R[x,y]

o2 = S

o2 : PolynomialRing

i3 : F=a*x^2+b*x*y+c*y^2

        2              2
o3 = a*x  + b*x*y + c*y

o3 : S

i4 : (M,C)=coefficients F

o4 = (| x2 xy y2 |, {2, 0} | a |)
                    {2, 0} | b |
                    {2, 0} | c |

o4 : Sequence

i5 : C -- should be in R but is instead in S, forcing pointless "lift"

o5 = {2, 0} | a |
     {2, 0} | b |
     {2, 0} | c |

             3      1
o5 : Matrix S  <-- S

tangentially related: #2490

[pzinn]

This issue is particularly bad with AssociativeAlgebras:

i1 : needsPackage"AssociativeAlgebras"

o1 = AssociativeAlgebras

o1 : Package

i2 : R=QQ<|x|>

o2 = R

o2 : FreeAlgebra

i3 : (last coefficients(1/3*x))_(0,0)

     1
o3 = -
     3

o3 : R

i4 : denominator oo
stdio:4:11:(3): error: no method found for applying denominator to:
                   1
     argument   :  - (of class R)
                   3

[pzinn]

In constrast, leadCoefficient does give the coefficient in the coefficient ring, as should be.