When will we get a dragon rider subclass?

Make it a series of feats with prerequisites that anyone can pick up. 

This would make a much more interesting world.

Different classes can ultimately gain different Dragons as friends/mounts. Never dominating the sapient Dragon. Other dragons would not like that, in fact they could view it as slavery.

You would have to have an insentient species just like other mounts. And nurf the mounts power. Breath weapons and flight for low level characters are very powerful.

A hundred ideas could be added to your world. But like others have said. Make sure the DM is not handing a player the cheat code.

The point about being non-sentient- or at least not previously extant beings who would have lives and goals outside of obeying your instructions- is covered by Drakewarden. The issue is that despite recent pushes for "setting neutral" content and a downgrade to the emphasis of lore, you can't meaningfully have a "real" dragon by the reckoning of most D&D players that also fits the mold of a subclass feature companion.

Can dragon-riders be good to explore underdark?

Maybe a dragonsteed-rider and this with climb speed could be useful to explore underground.