Mapping twelve letters onto a piano keyboard would then look something like this:
B D G I K
A C EF H J LA
Which means an A major scale in this notation would be ACEFHJLA, which is actually less intuitive than understanding the circle of fifths etc. and arriving at ABC#DEF#G#A (to use this universe's notation)
To +1 the "no objective foundation": browse music theory research a bit! There's a ton of caveats and poor-sounding fits and whatnot for literally every approach, and there's endless discussion of it. E.g. have a MinutePhysics take on how the common 12 note western scale falls apart: https://www.youtube.com/watch?v=1Hqm0dYKUx4
(maybe to clarify: there are objective aspects, in that sound is measurable. but there is nothing like a "grand unified theory" that covers all music, nor are roughly any of the popular ones internally consistent - it's far, far too varied for that, and physics often doesn't allow the desired consistency, causing more variety)
In your traditional system, if you want to play something a step up, you have to actually think about it; which notes will now become sharps, which won't, etc.
In my system (A though L, or more simply, 1 through 12), you simply add 2 to each note. It's easier to work about and isn't as rigidly defined by the culture it came from.
You just described https://en.wikipedia.org/wiki/Whole-tone_scale. Nothing stopping anyone from writing and playing music in such a scale, but it won't help you with nearly all the music you are likely to be familiar with.
Mapping twelve letters onto a piano keyboard would then look something like this:
Which means an A major scale in this notation would be ACEFHJLA, which is actually less intuitive than understanding the circle of fifths etc. and arriving at ABC#DEF#G#A (to use this universe's notation)