Master ADC here. Not being in competitive doesn't mean shes weak and it doesn't mean it's a problem with the passive. You're using a common red herring to make your point. Why does her not being in competitive automatically have to be a problem with her passive? You can literally harass/farm and still build passive procs in lanes, it's nothing like you're describing it. You even play the champ bro? I fucking doubt it. Her passive is actually very nice for bursting people with ult/volly/passive combo, shits damage is UNREAL! I love it.
Here's some real reasons Ashe isn't used in competitive
Ashe works good with hard carries in other roles or champs with a lot of CC to chain it. Meta has very few of those, usually tanky junglers. In soloq I break out the ashe when I have stuff like Yi jungle, amu jungle, Fiora top, and so on. Shes definitely not bad, my team already has DPS for days so I'm more than happy to support them. AD's are usually the most dmg on most teams lately, teams are using tons of aoe ap/ad cc like Liss, rumble, j4, gnar, and so on. Even back in S4 we had a lot of Ryze/Mao/Mundo and stuff, and low damage tanky supports.
Pro plays are more slow paced, slow itemization of qss/crucible/etc makes her ult less effective. Ever play D1+ Rank 5s? People play around cooldowns SO DAMN hard. Ashe ult can be nullified by a lot of items and in slow paced professional games you can make her ult a lot less effective, both by playing slow and baiting it out, and also by itemizing so it's stun is reduced, or you can cancel it entirely. Also QSS was buffed so that shit is easy to buy if you want it. Even Banshees counters Ashe's ult pretty damn hard.
Consider the fact pros constantly don't realize a champion is strong or weak for months. We've seen tons of NA players try to use Rengar with zero success, and we've seen pros ignore very strong champions for months/years. Teemo, Soraka, Twitch, Khazix, Shyvana, Riven mid, Ryze top, etc there's so many champions which were neglected in pro scene until someone actually broke it out and showed everyone else it was strong. Most recently is probably Janna support, Jayce mid during World's. Even now people still don't use Jayce mid and he's still insanely strong, by the end of S5 people might actually learn him. Play Janna a few weeks before the last World's and your team would have probably flamed you.
Ashe definitely can work into a lot of team comps though, I'm simply pointing out some weaknesses in it. You'd just have to use more damage/cc heavy based team, and avoid picking her into people who can itemize qss/banshees/mercs/hourglass/etc super easily. A lot of mid laners can also dodge her ult, like Fizz and LB, so you just have to be smart about when you pick her.
Ashe hasn't been in the meta for a while, but right now isn't the best time for her considering how almost everyone is runnign stuff like Sivir(spellshield counters her ult hard), liss(can dodge it + has two hourglasses), fizz(easy dodge), lb(easy dodge), j4(go ahead and ult a J4. HE WONT GIVE A DAMN), Lee(gl ulting a lee), qss nerf, etc etc. So yeah, if you wanna pick her you have to build around her. This means more damage else where, try to not give them super good gap closers like J4, and don't pick her into stuff that can passively negate her ult(like sivir or leblanc.)