You need to read the earlier discussion more carefully. The majority of the cards you give as examples are either attacks, in which case you are conflating two distinct types of P1 advantage, or allow for multiple gains/buys, which I have said from the beginning can grant P1 an advantage even with identical shuffling. The other cards you mention do not provide the advantage to P1 that you claim. Why would P1 buying a mint on his third turn provide P1 an advantage if P2 also buys a mint on his third turn?

I have given no examples of using attacks in any mathematically worked example. Further NONE of my analysis has looked at the impact of playing the attack, merely who gets there first or who wins the split. For instance with Forge I have not even looked at its gain; I literally leave it as a breakpoint and analyze who gets there first (literally if all the forge did was forge curses & coppers into coppers my analysis STILL holds) . This "critique" is spurious and, frankly, I am disappointed that you could not argue without making it.

As far as mint goes. Let's say that both players open chancellor/mining village with a goal of minting to big money (which for whatever reason is dominant here). To simplify analysis the chancellor will always hit for both players (no luck advantage there). Now suppose P1 hits mint on T2 & chancellors. Then he has trashed 3 (4) coppers & has a mint. This will happen X% of the time. Now P2 may or may not hit mint & chancellor as well. He, likewise has X% odds. (X^2)% of the time we will have a degenerate match. However, X(1-X)% of the time, P1 will be playing with a short deck and have a strong advantage. P2 will be in the reverse situation only (1-X)^2*X% of the time. Hence the odds of P2 having an extra turn with a thin deck are much lower than those for P1.

The scenarios are these:

Most often: P1 & P2 buy mints on T3 or P1 misses & P2 misses (no advantage)

Next most often: P1 buys a mint on T3 & P2 misses on T3 (advantage P1)

Next most often: P1 misses a mint on T3, buys it on T4 & P2 gets a mint on T3 (no advantage)

Least often: P1 misses a mint on T3 & T4 and P2 buys a mint on T3 (advantage P2)

Now you correctly note that for many decks purchases on T3 or T4 are equivalent, this is true, but for the purposes of this discussion that can be largely abstracted by saying what are the odds that P1 hits something on T3 or T4 and what are the odds that P2 hits something on T3 or T4. We could run the entire analysis again without the lucky chancellor condition, but the net effect is this degeneracy diminishes, but does not eliminate the first player advantage from having a probable extra turn.

This is a matter of statistics. We could, in theory, derive correlation coefficients or do an ANOVA regression to determine how much the variance in player achievement is driven by turn position. "Chalking it up to luck" is bad mathematics and incorrect. Some of the variation in player achievement comes from skill, some comes from random card distribution, and some comes from position.

I believe that skill > card distribution > position; but these are three separate quantities which can be observed in the long run. Further I believe that if skill and card distribution are held equal, then position would be determining in all but the simplest engines where the final determinant - the tie breaker rule - would dominate.