By this definition, Bardet, Sosa, Kuss, Simon Yates, Vingegaard, Vlasov... aren't climbers. A lot of riders can drop them on longer climbs.
During the Euros, he dropped every climber in the race (Almeida, Bardet, Pogacar, Sivakov, Mollema...) on a 3.5km climb. The only rider he couldn't drop was a "sprinter".
A recurring theme, he needs to prove again and again what he can do in order to be sure, and the first moment he falters, it's proof he can't do it. He drops everybody in Burgos and Algarve, but that doesn't prove anything. He's "only 5th" in Emilia (ahead of Vingegaard, Kruijswijk, Martin, Mollema, Quintana, Pozzovivo...) after leading out his teammate in the final and finishes poorly in the last race of the season (yet ahead of known climbers) and he gets demoted to being in trouble on anything longer than 3k climbs. Imagine going into the Pogacar thread and question his climbing ability based on his Euro and Emilia performances.
It's all very tiresome really. If anything, what he did in the Giro, knowing he was on a downward trajectory of form, was reassuring imho. He was still in the top 10 after 14 stages and even finished the Zoncolan stage "only" 1m30s from Bernal. Even if you want to ignore the circumstances regarding his injury, it being his first race in 9 months and his lack of base form, a rider his age that finishes so close on such a stage would be seen as a future climbing force to be reckoned with. But in Evenepoel's case, let's use it as proof he's not a climber.
He may not be a pure climber, how we imagine a Contador or Quintana. But i think you'll have a hard time finding 10 riders with better climbing performances than him at that age in the past decade.