You say,
Again, it would seem like Maxima fails miserably... and I'm not even
taking a very difficult case!
Why not try some function f(x)= stuff,
where f is sin, cos, linear, quadratic, cubic, quartic.
instead of f(x)=g(x) where neither f nor g
has a single-valued inverse.
Isolating a variable on one side of the equation by doing
(generally) illegal transformations is not something
people tend to put into algorithms that are supposed to
do correct mathematics.
Expecting such a system to perform your way (without
complex numbers) would require Maxima to fail to
solve x^2+1=0. Whle programs could be written
to do this, you may have to do it yourself.
The commercial Macsyma does some of the problems
you indicate, by the way.
You might also learn to use the tools like expand, factor,
ratsimp, trigsimp, etc. Solve may not
do what you want, but there are other tricks available.
If you want to write a program that mimics a naive
math student and uses only the techniques you have taught,
then that can be done.