Welcome, Petecon! We recently began the renovation of an existing gunite pool, and explored the option of reshaping the pool to a simple rectangle with autocover. It was cost prohibitive for us, and essentially the difference was between a $35K basic reno and a $60K extreme reno. I would guess the allocation was about $10K for the autocover portion of the bid, so about a $15K-20K cost just in reshaping the pool. That would not account for the amount that it would cost to convert vinyl to gunite, but might help you gauge a little bit for the reshape. In our case, it was sort of "once you have the gunite truck out, you might as well get the tanning ledge, bench, swimout, etc. you might want" as there was somewhat a baseline cost to just get started and every little thing didn't double the price once they were there.
In our area, even the basic reno of $35K for our large pool is comparable in price to building a brand new smaller pool. Just a case of more square footage and more materials making a much larger impact. In our case, we could probably build a smaller pool (if we had the room) for about the same price or less, but not actually demo the existing pool within that budget. In hindsight, I wonder if buying a house with no pool and building a smaller pool someday would have been more economical, but c'est la vie!
In your case, I would suggest starting to get some bids, and ask builders for two bids - a basic and "extreme" remodel and see where you end up. I wish I had required a standardized list of our reno must-haves to have builders use so I could compare all apples to apples, but as I met with each one, new issues or considerations were raised or ironed out so the bids all vary slightly in scope, even though I gave them all the same sketch and list of items to work from.
If you do not have the funds to completely change from vinyl to gunite, I think one thing that would really change the look of your pool would be to change from the cantilevered coping to a regular coping (like the bluestone you mentioned). In my mind, that is more consistent with a gunite "look" so might help you get the look you envision. One of my favorite styles of pool is bluestone coping with grass right up to it (a very Cape Cod style, which I think is what you described) but my husband nixed it for mowing purposes. We even looked at artificial grass, which would have solved the mowing issue but in our climate, a perpetually green lawn just isn't natural so we went back to a concrete deck.