Everyone agrees that unassailable proof (analysis of every possible move from the starting position to the ending tablebase) is impossible to achieve, so we are all nit-picking the amount of evidence each person feels is sufficient to satisfy them that the game is a draw.
No they don't.
Everyone agrees that individual analysis of every possible move from the starting position to the ending tablebase is impossible. Not everybody (me) agrees that it's necessary.
If a forced sequence is found for one side that results in mate before the endgame tablebase is reached or forced sequences are found for each side that result in stalemate or (under competition rules) expiry of the 50 move rule or a triple repetition before the endgame is reached then analysis of every possible move from the starting position to the ending tablebase is not necessary. If there is a relatively short forced mate or a relatively short pair of forced draws a forward search finding these may be possible in practicable time.
Neither can you discount the possibility that a mathematical proof not based on exhaustive search could come up with an answer. For example, you were probably taught early on a method for mating with K+R against K which was arrived at without any use of search at all. A full search would be feasible if the starting position is a relatively short mate, for example.
Such a proof has not progressed very far yet, but give @tygxc 5 years and 7 maids with 7 mops and he could probably at least prove he's a teapot.
I read the first couple of paragraphs of MAR's reply and reading further was unnecessary in this context because I completely agree with MAR that a complete tree leading to a tablebase is unnecessary for a proof. Should be obvious but never mind. It's also at the moment impossible but let's not let that put us off.
OK then.
Bubber off!