Q:  So my question is: is it realistic to hope for a dramatic skill improvement as a result of obtaining a keyboard, or am I a victim of capitalism foolishly hoping that buying something will fix my problems?
A: it's going to be impossible to learn music without an instrument. even if you only sing, you're probably never gonna feel the power of music until you can play with some chords too. which keyboards tend to enable, quite easily in a friendly manner