If you agree that chess can be solved in this way, then you must also agree that there is no luck in the game of chess. In the above scenario, we did not change the game of chess in order to remove luck. We only changed outside factors (the intelligence of an engine).
The "luck" involved in chess has nothing to do with chess at all. It is like saying you need to be "lucky" to get 100% on a math test. There is no luck involved in math (2 + 2 always equals 4), however, someone can accidentally miscalculate an answer. Would you say there is luck involved in math similar to how there is luck in chess?
There is no proof needed, logic and common sense is enough. Let me help you.
The strongest engines in the world make mistakes for the reasons I named in a previous post; Since the engine is unable to calculate all variations till the end, there is always information missing, based on which the engine has to make a move.
So we know stockfish makes imperfect moves. Take any imperfect stockfish move for an example. Now in the same position we use a random number generator to determine a move. The rng can hit any available move in the position, so we know chances for the rng to hit a better move than stockfish exist and they can be calculated.
Here we have a situation where stockfish obviously applied more skill and calculation power but got a worse outcome than rng applying 0 skill. So skill isnt a factor here, what is? (Hint: its luck)
I see where you are coming from. What if you instead use an engine that CAN calculate all possible variations until the end (assuming one existed)?
In this scenario there is no luck in this particular way. Because the random variable is created by information that is missing while making a decision. But if an engine or human could calculate till the end, chess would be solved and there is no game.
Until that, luck exists as I explained (and in other ways too).