Your third paragraph contradicts the second.
It's very possible to not know or care about equivalence, provided you never change camera format. But if you do, and want to know about the differences you're seeing, then check the DPReview link posted above. You'll want to know about how the crop factor affects field of view for a start, and also how depth of field changes, also by the crop factor. There's a lot more to it than that though and the crop factor, or crop factor squared, comes up time and again.
That DPReview link doesn't go into how equivalence affects lens performance, but it most certainly does. Better sharpness is probably the number one reason folks switch from APS-C to full-frame. The crop factor is part of the calculation there too.
Why are smaller formats more noisy
Bigger has undoubtedly always been better... but there is no need to dress it up with some magical "equivalence "
Bigger can mean less enlargement is necessary. Or that you need to open up the aperture less to reduce the depth of field. Or can refer to many other factors with a mathematical relationship. Unfortunately Bigger has also always meant chunkier and heavier and perhaps less convenient.
In real life we chose a camera of a particular format because it is "convenient, cost effective, does what you want it to do, and is good enough".
My little Fuji X20 is good enough at the sizes I need and for what I need it for. Were it not, I would use or buy something different.
I never used 35mm professionally, except for the very few occasions that a client wanted 35mm slides. Howevr I rarely used anything else for my happy snaps after the Dslr became "good enough".
Format sizes never have an equivalence of usefulness, one always stand out as the better option.
I under stand all the mathmatics and theory behind crop factors, focal length, apertures, magnification, exposure, fields of view, etc. to the extent that I now rarely have to think about them to make my choices.
"Equivalence " does not effect lens performance. because red herrings rarely do much at all, but magnification to the final image certaily does.
Hence... if you want big and good ...Start big.
Equivalence is an confused observation not science. It is a very poor way of thinking about relationships, it seems to confuse and blind with science, in equal measure, rather than give accurate explanations that people can actually understand.
When Photographers used many and varied formats, neither crop factors nor equivalence were ever used to explain anything. They seem to be a totally unnecessary product of the digital age.