I can think of a few reasons why Weavile is good in Durant.
Using Sneasel provides an extra starter, so you won't be getting mulligan after mulligan after mulligan. Though you may think mulligans are good, they are actually bad because it gives your opponent more cards they can use to get an attacker.
The T1 4 Durant is more important than not giving them an extra 1-2 cards. Sneasel gets in the way of that. Besides, mulligans can be good. They have 47 cards in their deck after setup, mulligan twice and you took a turn off of the overall mill turns. However, they usually take these two cards off anyway shuffle-drawing into an attacker.
By not getting mulligans, though there may be one or two extra cards in your opponents deck, denying your opponent the chance to get an attacker up by not giving, or at least trying not to give your opponent extra cards through mulligans. Sometimes a player can have a really good hand, but no basic. By giving two-three spaces for Sneasel, it is more likely that you will get a basic while you have a good hand. Also, there is no harm starting with Sneasel because Sneasel has a free retreat and a reasonable 80 HP.
Sometimes a player has a really bad hand, but no basics. Add in Sneasel, and you could get stuck with a bad hand. The cards in your mulligan hand are ultimately irrelevant; saying adding Sneasel creates better starts is not only just superstition, but it's also not true. Sneasel does not have 80HP.
Disruption is key. I have discarded important cards using Weavile in my Durant deck. I don't use any search cards for Weavile either, but eventually using a hand refresher, you'll get a Weavile.
Weavile is best when used before your opponent is setup (turn two.) As good as hand disruption is, it's not going to help you if your opponent already has a bunch of attackers up.
Sneasel will not be Catcher'd up and ko'd also. The main focus against Durant is to knockout the Durants so your opponent doesn't mill as much. A Pokémon Catcher will be used to knockout benched Durant instead of the Sneasel.
The idea of any deck when going against Durant is to take six prizes. If they're using Donphan or Yanmega and can't OHKO an active Durant, they will KO the Sneasel. It speeds everything up by a turn, and chances are you have the Revive anyway.
Weavile discards potentially important cards such as possible attackers, energies needed for attacking, or even cards that are necessary in certain situations. Key discards can stall your opponent from attacking, giving one more turn for Durant to mill. If nothing else, Weavile will make one less card go back into your opponent's deck after they use a hand refresher such as PONT, or if the Durant player uses N.
True, although you'd hopefully remove the hand refresh via Weavile in the first place.
Why not? It is a four-eight cards tech (depending on the Weavile line you run, I do not recommend 4-4 Weavile) with potential to disrupt and stall your opponent. Sneasel is a good starter, and Weavile isn't Pokémon Catcher bait because of it free retreat. With free retreating and Sneasek as a starter, Durant could actually gain consistency. Lost Remover and Crushing Hammer are used to disrupt your opponent, and Weavile shoud be used too.
-4 consistency cards. +2-2 Weavile. Mind explaining how that has ANY potential to add to consistency? Durant is a better starter than Sneasel. And if you're going to go the, "but I wouldn't drop consistency" route, what cards would you drop? Weavile is Catcher+KO bait, since it has a bad weakness, low-HP basic, and lower HP than a Evolited Durant. I don't necessairly directly disagree with Weavile in the deck, but these are not the reasons to include it. The only reason is the second-to-last paragraph you had in here.