Can you tell me what his argument is for that?
An on-demand stun and her own ability to peel. She trades off raw damage for the ability to prevent anyone from reaching her outright. Honestly, the only thing she lacks compared to, say, Ezreal is an on-demand escape in the form of a skill that creates distance. Ashe controls the space in front of her very well.
Or just this:
There's more to being an ADC than how quickly you run away.
Think about this: Renekton could theoretically use no abilities until he got into range of you. Same with Zed. Same with their jungle Amumu.
You're Ashe. You can Volley and ulti from outside of their engagement ranges.
Do your opponents: run at you and never catch you as you run away, making it a 2v4 fight? Do they say screw it and use abilities on the rest of your team, which now allows you to enter the fight fully because their CC is down? Or do they just send one guy after you, whom you kite with Frost Shot?
Now granted, this is a somewhat specific circumstance, but I'm trying to inform that there's more to team fighting as an ADC than just walking up to auto attack range and then going, "OH **** IT'S IRELIA" and trying to run away. Compared to Vayne, Ashe has a lot of things she can do from safety. This helps Ashe be a "safe AD Carry"
The game is not ADC vs. the world. The game is 5v5. There are teams. There are 5 champions on each side who have varying levels of, "Give a ****" if they're getting hit or not.
The enemy team's threats don't get to fight optimally and threaten an AD Carry. Zed doesn't get to Death Mark everyone on the team. He either sits there toe-to-toe with Jarvan, or he sits in the back hoping he can find a way in toward the back line. If Zed sits there throwing around basic attacks, Ryze and Jarvan eat him for lunch in about 3 seconds. Now Ashe gets to fight! My front line had to tank for 3 seconds, and I was able to still ECA and Volley from the back line. Zed did nothing.
Or what if Zed decides to sit in the back? Well, I'm Ashe! I am 600 range away from your front line. I can see Zed coming, Volley from 1000 range, go, "Nope!" and get out of there before he ults me. Also he's slowed, and, if their cooldowns are still available, J4 and Ryze eat him up.
Now, if I'm Vayne it's different. I'm either in the fight or I'm not. I don't get to bait Zed. I have to outplay Zed. If I'm fighting, he's in reach of me. If he threatens, I leave the fight entirely. Now maybe I'm far enough ahead, or a good enough player that I win the 1v1 or escape the duel successfully and leave him overextended, but the point still remains that Vayne isn't exactly safe in a team fight.