From “Things”, I gather that you might be referring to their “Mail to Things” setting.
Which works because there’s “Things cloud”: Some kind of not very well detailed service sitting somewhere in “the Cloud” doing something magical (and hopefully respecting your privacy). BTW:
With Things Cloud, all your network traffic is securely encrypted using the latest SSL technology, ensuring that everything you do is for your eyes only.
(quoted from their website) That means they do transport encryption. Not a single word on data at rest, i.e. in their cloud. Not a single word on where the data is located (US? Europe? China?). Nothing on the cloud provider (Google? AWS? Azure? Apple?). Hopefully, not the latter. This lack of detail would make me want to avoid their service.
Yes: DT does not offer a cloud service. Which I, for one, do not regret. But I remember a discussion on that here in the last several months, so you might also want to search in the forum (which could prove a bit difficult given the ubiquity of possible search terms).
Given that the OS for desktop and mobile are vastly different (and probably will remain so for some time), not everything possible on one platform can be implemented on the other (there is no “swipe up” to terminate an app on the desktop, for example). Then there’s the question of real estate and limited resources – you do not have swap space on a mobile device as you do on a desktop. Heck, you don’t even have a real file system in iOS.
Catering to unrealistic expectations is pointless, in my mind.
Well, that again excludes all other mail clients. It also requires Mail to sync its state and content across devices. Unless you mean an IMAP folder (of which you’d need one per account). Which again leads back to my earlier point: DT would need a kind of IMAP client to work across platforms and around clients.
What you really (seem to) want is
a cloud service that integrates with DT and DTTG in that it magically keeps your data Somewhere™ and offers a (secure!) API to talk with it, including sending e-mails (or whatever documents) to these services and then do Something™ with them.
Something like “iDEVONthink.” While I see your point: That’s not where DEVONtechnology seems to be coming from. The product has been placed as a single-user, two-machine app (apart from the server, that is). It was not conceived (apart from the server, again) to be running 24/7 and waiting for requests to arrive.
Even if there were an iDT software, it would probably come with a very different price tag, rather with a subscription model: ingress/outgress as well as data at rest have to be paid for by Someone™.
In my mind, it would be a lot easier for everybody if Apple implemented a “share” menu in their Mail program. Well, one can dream.