I've started groups as hunter and rogue. They take longer to fill than the ones I start as tank, but they've never taken more than 20 minutes or so. Helps to include your experience level in group description, assuming that's a positive.
As others have pointed out, the best solution is either rolling a tank or heals, or getting to know tank or heals who trust you and enjoy running with you. I have hunter and druid friends I know I can count on; they are first ones I look for when I want to start a group.
As long as more players choose to play DPS than the other specs, there will be a shortage of those other specs.
It does turn around a bit when it comes to raiding. Only 2 tank spots compared to 5+ DPS spots. RDPS, in particular, can usually find a spot right away.
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Yeah, tbh, when I pug DPS for anything that isn't critical, I usually just take the first acceptable I-lvls I see in queue. I get a little pickier for good keystones, since I don't want to risk depleting them. But more than once I've accepted one app, only to see someone better - like you - a split second too late.
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^^^^Yep. This is closer to my experience. I suspect OP is frustrated and exaggerating.