Well Turalyon is a legendary hero of the Alliance, everyone respects him. Turalyon was the original founder of the Alliance of Lordaeron actually, the one who rallied the Seven Kingdoms and convinced them to ally, and he was also Lothar's successor, who finished what he started. He defeated the Horde, rebuilt Stormwind (keeping its throne safe until Varian was able to rule), and then led the expedition to Outlands to finish the Horde remnants.
Turalyon is for all intents and purposes the biggest hero in the Alliance, perhaps second only to Varian himself. Depending on how much weight you give to his final sacrifice at Broken Shore.
Regardless I can see most of the Alliance backing him due to his reputation, and for lack of alternative really.
And no doubt Alleria would support him...
Since that would make her High Queen technically
Also it should be noted that Yes, Velen, Greymane, and Jaina are like family to Anduin, and it is precisely for that reason that they might realize going back to the throne might be bad for Anduin's health. He needs a vacation after getting mind-tortured by Satan for 2 years.
And really I think that most leaders who were there in the Second War (Greymane, Council of Three Hammers, Mekkatorque, Danath) would fully support Turalyon. Velen and Tyrande would support Turalyon if anything because he's ancient like they are, and is a proven defender of the Cosmos.
Turalyon fought the Legion for thousands of years, certainly the Draenei and Night elves would reward that.