There is a huge misunderstanding of the meta / prevalence of AoE specs in the dungeons.
Imagine that dungeon XP was heavily nerfed. Many people would still do some dungeons once or twice for the quests and or some sick drops. Including mages and locks.
Dungeon XP is significant howerver. So the levelling meta is to AoE dungeons. Instead of doing Scarlet Monastery Cathedral once or twice, for the cloth helm and quests, a mage will do it 20 times, or even more. So due to the meta, mages and locks do 10 times more dungeons than usual.
That means that when you are considering how difficult it is for non-meta specs to get into the dungeons, you need to ignore 9/10 of the AoE groups. Had the mages not been the meta, they would not make the group you are rejected into to start with.
I have done SM AoE grinding from 30-44 as a priest healer. I have grouped up almost exclusively with mages/locks. If a ret would join, I’d almost definitely leave after 1 run. Because a ret in a spelcleave completely destroys the purpose of it: pure XP/hour. I’ve also grinded ZF GY for a few levels. Mages exclusively, lock only once. But I’ve also done 2 ZF full runs, advertised as such. In those runs, I did not have a mage. The groups had rogues, hunters, prot pallies. In one of those runs, I actually DPSed as shadow. There were no questions asked about shadow being a memy spec. There was a post, LFM DPS ZF full run, I said, I can DPS as shadow, instant invite.
I did 2 maraudon full runs, no mages again. But there were hunters, rogues, a DPS warrior, a lock. The groups were filled up relatively quickly, i joined one and started the other.
Getting into quest run dungeons as a non-mage is as difficult as getting into them as a mage.
Dungeon AoE runs are a completely different beast, they are not dungeon runs in the standard sense and should be ignored by people who only aspire to do the dungeon a few times.
That said, your problem is not that you are not the meta spec, you are the WORST spec in the game. Everyone knows you are a bad spec, including you. Of course people will reject you as a DPS. I’ve seen rogues, hunters, warrior DPS, but I’ve seen 1 ret in total in my dungeon runs, and 0 oomkins. And that’s fair. You knew you were not a reasonable spec when you rolled, and expecting people to carry you even though you can either heal or tank decently, is your problem, and not one of the community.