Absolutely. And that's what you trade, just what you say you see and what everyone else sees too. There are no hidden cycles, waves or magic numbers. You can fit Elliott Waves to as many price formations as you can any other artificial technical construct, model or abstraction. The problem is, those price formations you can't fit them to outnumber those you can by many orders of magnitude.
You need to be stripping away everything that isn't the bare price action, just as you suggest.
I agree with stripping away the superfluous. But adherence to the KISS principle for me does not lead to a bare price chart. Tools are useful and relate to a person's knowledge, skills, and experience. A chart of bare price bars on any of my display space means it hasn't been formatted and no work has been done on it. I do burn a lot of midnight oil around rollover time updating charts of the new front month contract.
I find the EWP interesting. R.N. Elliott saw what everyone else saw. His spark of genius was inventing a framework to overlay on top of what he saw. As you point out, trying to fit every market movement into the framework raises practical problems. Sometimes the fit is very sloppy and backtracking is required to clean it up. In any case, theoretical perfectionism offers little utility in trading.
From a practical viewpoint, counting is basic to trading. The simple reason is that the now and immediate future relate to the immediate past and are informed by it. Whether or not 3 is a magic number I do not know, but 3 is the unit of construction for a trend. Although it's no secret that the market oscillates, to the ignorant everything is hidden in broad daylight.