Thinking of them as a derivative of a major scale is a short cut that ends up slowing you down dramatically. You don't want to have to have an extra mental step of calculating back to the related major scale when you need to think of a different scale and use it.
Being able to think of a floating specific set of notes other than major, changing the set as needed is a crucial foundational tool for improvisation over changes with even a little bit of complexity to them. You may as well get started on that sooner than later.
One of the interesting things about this is that by thinking of each mode as the unique scale that it is while you are playing them, you tend to retain the sound of that scale while improvising. Whereas if you mentally relate them to major you tend to end up sounding major. This is no small point, and a lot of people have conceptual trouble with the idea that the same set of notes can take on all of these different flavors by emphasis and context. The intervals you choose and the notes you favor reveal your thinking.
^^ Yes. The common insistence on tying modes permanently to the major scale can actually hinder understanding of what they are best used for IMO. It can do this in two ways.
1. First, it can overcomplicate standard diatonic harmony without adding any explanatory power at all. I see many cases of folks believing that when they play a standard unaltered diatonic progression in, say, C Am7 Dm7 G7 in C, they're actually cycling through four different modes. There's no real need for this. That's not the most useful or elegant or even correct way to think about it. Really, that kind of progression is just moving harmonically through a single key. Now, there do exist musical segments where you really
are changing your harmonic space with virtually every chord, but the types of standard diatonic functional progressions usually cited as examples are not that kind of thing.
2. Second, it can hinder the recognition of when you actually
are playing in a mode. Some music--eg, certain folk or ethnic music, as well as jazz--is probably better understood as based on modes rather than classic functional harmony. If we're convinced that, say, D dorian is inextricably linked to C major, we might think we're "really" playing in C major when we're actually playing in D dorian. This would probably be kind of counterproductive.
This is why I usually suggest that when it comes time to address modes, people learn them from the same starting note--C ionian, C dorian, C lydian, etc. IMO that's a better way of getting a sense of how the sounds of the modes differ.