The problem with including price is that it creates universal edge cases about three piling that don't care at all about the card mechanics.
This is why I made a distinction between "strictly better" and "strictly better effect".
Purchase price is only part of the concern with cost. With a Haggler in play and having bought a colony "price" is meaningless, but there are times I want any $4 over any possible $5; most obviously with Forge. Price has to be ignored or I can edge case any
cards of differing price. Woodcutters vs Festival is literally the same effects +extra actions, but if we concern ourselves with price (even ignoring purchase price), there are times where Woodcutters beat Festival.
This description is a bit convoluted. Garden->SR or vice versa says nothing about price because they both cost $4. Your $7+ example makes no sense because the $8 card can fill in for the $7 card just as well. You can switch to using Upgrade though, but that's certainly something worth considering that can make you prefer a $7 card over a card at another price point.
They are convoluted to avoid easy edge-casing. The point being cards like Remake, Forge, Upgrade, Governor, Apprentice, Graverobber, Rogue, Knights, Sab, etc. all can create edge cases where point values make things difficult.
You're abusing the term "strictly", which should mean "in every situation". If A is only better than B in some situations, it isn't strictly better. It's merely better in that situation. And that's fine. Absolutely price can make some cards better in a specific situation. The easiest example for this is Forge; we don't need all the hyper-specific examples you gave above.
Not at all. If A is strictly better than B, then A > B holds in all possible situations. Festival > Woodcutters is true for all non-nearly universal edge cases (e.g. Possession wants crappy cards in your deck, you want to pass something worse with Masq, $3 + Peddler = colony with Forge). VP card > Curse actually only holds true for a subset of all possible game states; first you have to ignore game states where grabbing a curse can be good of its own right (Ambassador/Fairgrounds/Mountebank, etc.). Then you need to get rid of times where 3-piling makes curse buying the right move. Then you need to get rid of cases where you care only about the fact that curse is 0 cost card (e.g. an Embargoed curse pile makes curse "buys" better for Trader). Then you need to get out into the weeds where stuff cares about the fact that curses are worth
0 (e.g. Remake, Upgrade, etc. when Poor house needs to be gained and the copper pile is empty).
Any rubric that allows us to say that Woodcutters > Festival or Curse > anything else is likely just showing us near universal edge cases.
Ignoring worth has its own problems, most notably Peddler has major issues where stuff that has some additional effect is "strictly better", but Peddler can be bought in bulk or bought cheap but trashed high, but the answer is either "nothing is strictly better than anything else" or we adopt some basic guidelines (which due to the richness of Dominion need to be pretty long) to ignore the near universal edge cases.