Some vocal techniques - like belting and the classical voices - require the actual development of musculature, both to hold a shape (for high pitches) and to be able to stabilise shapes. That's not going to happen overnight, so you have to spend sometime sounding horrible before you can do those well. That doesn't mean it's impossible - it's like lifting weights, anyone can improve it.
However, that's not the techniques used in the majority of popular singing. You absolutely can sound drastically different very quickly. This is simply because most people don't use most of the degrees of freedom that the voice has. If you look at the Estill method reference material, which concentrates on how the vocal system features actually operate to produce different vocal effects, they identify around 14 degrees of freedom.Some discrete, but a lot of them continuous. It is very common for someone who thinks that they have a terrible voice to take a few lessons and find that they can sound much better. It's just that, as another commenter pointed out, you can't see inside your own mouth and throat. You can't see that you're always holding them in a certain way.
For example, some women habitually sound shrill, usually because they have had to develop a penetrating voice. This is often true of teachers, who have to be able to make themselves heard over a bunch of shrill children (and aren't able to use the option of a deep bass). This is a vocal technique (twang) but it can also become habitual,to the point where they don't think they can do anything else.