>The higher-end work – [...] – is still around, and it's become much more specialised
The specialisation means that work for "web designers" has declined IMO.
I used to do sites for local charities and SMEs. They tend now to use Wix or WordPress with a template, something along those lines.
Those who can afford something more bespoke have much greater expectations and web design as a field has grown. Just keeping up with the pace of change of browser tech is hard now, never mind everything else that goes in to a "simple" webpage - one needs a team to do it properly IMO. So the generalists, as I once was, are really not in demand anymore.
The web designer has been killed off by commoditisation and specialisation and there increased expectation and possibilities that are now part of the everyday web.