I'm not really saying you get closure. I'm saying that the story is done. At least, for the Inq, certainly in DAI anyway. He's a set up for a future wild card (since Flemeth no longer fills that role).
As a character clearly they intend to do a lot more with him. As a person then yeah he did just straight up leave you, especially since the things mentioned in the podcast are never officially voiced.
But as a character arc, as written by Bioware specifically for DAI, the relevance Inqi and Solas have to each other is likely now a 'Keep decision'. It will have bearing in the future, but I don't think you should expect a wealth of content, because this was about setting him up.
It may just be like Hawke's LI is. You talk to Hawke, ask why they're alone, and they'll mention their LI and what they're doing, as well as why the LI didn't join them.
So, Inqi may mention Solas, or find a trace of him, or maybe he'd sent a letter in a later expansion/sequel, but I do not think he has more story specifically. People asking to 'finish' the story - it IS finished. It had a sequel hook ending, sure, but that's the story as it stands.
Also, as mentioned before, Solas romance is literally there because it is so delightfully depressing. The lack of closure contributes hugely to that.
Solas being a future Flemeth like figure does not need to exclude him from future DAI content, there's no reason why it should... However, I don't think that is going to be the case...
I didn't get any feeling of the story being done at all, there is plenty of scope for relevant DLC. The blight for example - is a concern for everyone including Solas - few things get him as animated as his dislike of the Wardens, and he is clearly concerned about their plans to combat the blight). We have just created a massive and far reaching organisation and the plans and methods of the Wardens have been questioned. Should the inquisition not be concerned by the idea that killing all the old gods might not end the blight? That's a huge concern and getting to the bottom of what the blight is and finding a way to stop it is an obvious priority for the inquisition. Ignoring the blight is akin to us ignoring all the global environmental and climate concerns while we fight wars and worry about interest rates... I want the inquisition to be better than that.
In the break up conversations it is clear that he's not breaking up because he doesn't care, it is clear that there is something else going on. He will not tell the inquisitor that he doesn't care, when pushed to do so he says that he can't, so it is never as simple as him just leaving.
Solas was made a LI to demonstrate how sad his situation/story is, this helps underline that he's a more sympathetic character. It would have been hard to evoke this so well without the heartbreak, that does not mean it was done to make players sad as such - that is just the end product.
'delightfully depressing'? for me there is no such thing... it was beautifully written and the cinematics were fantastic but delightful? nope. If the story gets some proper closure in DLC, it may become delightful but if it doesn't get any closure and all we get is what you suggest it will just become annoying.
I know many are hoping that the Inquisitor will be the protagonist in DA4, but I am not one of them. I am really hoping for a new protagonist and ALL new companions. I don't even want NPCs like Harding to become companions, I want characters I have never met before. I'll be happy for some cameos, but really, a clean break would be nice and I'm sure I read that they will be moving away from Ferelden, Orlais etc so that will be a good opportunity for change.
I have no interest in hanging on a major character cliffhanger for 3 years, I learnt that lesson after the wait for ME3. I really hope that doesn't happen.