Was it possible for Albus Dumbledore to disarm and defeat Draco Malfoy without his wand?
Probably; if Garrick Ollivander was even partially accurate about the use of wands and magic, then if anyone could do it, it would be Dumbledore.
“Oh yes, if you are any wizard at all you will be able to channel your magic through almost any instrument. The best results, however, must always come where there is the strongest affinity between wizard and wand…” (Ollivander to Harry Potter at Shell Cottage) Harry Potter and the Deathly Hallows - J.K. Rowling
Even if a wand that chooses the wizard is the best way to go, someone of Dumbledore’s ilk should have been completely capable, had Albus been in a stronger condition.
Dumbledore could make himself invisible without a Cloak of Invisibility, could see through the most powerful one of those ever created, could deduce when an equally capable wizard had transfigured himself into a piece of furniture and was the only wizard the other greatest sorcerer who ever lived feared.
I suspect back in the day, Albus could have turned Malfoy into a paisley Blast-Ended Skrewt with green polka dots had the late headmaster had wanted.
Unfortunately, Albus made that little faux pas of thinking he would be perfectly fine drinking the water from the fountain in the Cave by the Sea.
I’ve always suspected the curse of Gaunt’s Ring had been much more painful than Dumbledore ever let on. It seems like the pain must have been so unbearable it began to affect Dumbledore’s logic.
Otherwise, why put himself in such a precarious position, where he might have died from willfully ingested poison or what ended up actually happening; becoming so weakened that he couldn’t properly defend Hogwarts at the moment it was in the most need of his ability.