Starbase vendors. I can go to the EC vendor and buy stuff using EC and sell stuff using EC.

By the same logic, the DIL vendor right next to him sells me stuff, and in equality I should be able to sell my gear to him in kind for dilithium!!.

If EC and DIL are both "primary currency" in the game, then shouldn't we be able to RECEIVE them as such?

The answer(s) is(are) rather simple:

The Dil Vendor sells "above and beyond normal gear", The EC vendor does not.

The EC is a currency that is subject to extreme inflation... Several people (including me) are in no short supply of EC, and will likely never become short of it.

Dil on the other hand, is a in-game currency that (sort of) has a real life value, as you can trade it into a currency based on real money.
That is also why the Dil is gated to 8000 pr. day, so that no matter how much dil ore you get, the market will always remain under relative control, and will never become subject to the same rate of inflation, that EC has had since day 4.

While you could argue being able to sell back items you bought with dilithium, there's no way dilithium based vendors will ever buy absolutely anything from you.

Maybe if purchased items could be tagged as purchased with dilithium, and purchased by you, so that if you return to the vendor with items that you purchased with dilithium, you could trade them back in for at least some dilithium in return. Basically only you could sell back items that you bought, anything sent via mail or sold on the exchange would be sellable for energy credits only.

This would help at least to reduce the cost of swapping high level item sets, without being exploitable by anyone simply grinding easy to obtain items.

Can you sell back Dilithium-obtained gear to a normal vendor for EC's?

(Never tried it, the answer is probably 'no', but thought I'd ask...)

Depends on the gear but sometimes. Most items are simply spawned with preset stats.