I’m biased, I used to work at GDS.
The lack of a ‘login’ button is a huge issue IMHO, but the main blocker to it isn’t the tech teams, its at the ministerial/senior civil service level.
The problem isn’t the button itself but the infrastructure behind it. I.e. it requires a single database of users (i.e. a national register of citizens).
This needs doing, but politicians come out in hives because it involves setting up a system similar to national ID cards (politically difficult). Senior civil servants dislike it because its a question of which department owns it, if its GDS its the cabinet office, and that means the Home Office, DWP, HMRC surrendering some control (i.e. being increasingly dependent on external systems), which is something departments seem to dislike.
It’d save millions (billions probably) in the long run, but there isn’t the political will to do it. It would need an influential cabinet minister to push it through and would take 3-5 years to get properly embedded.