Yes and no. Of course Mobile First is still relevant.
It's not about pixels. It's not even just about screen size. It's about context and location and attention and convenience and portability and constraints and content and strategy and organization and relevance and lifestyle and much more.
Responsive design can end up being nothing more than a new coat of paint. After a while the flakiness underneath is bound to show through. A responsive website doesn't automatically mean you're all done on mobile. It helps a bit for sure, and it's part of the process, but if you put lipstick on a pig, it's still a pig - a responsive website does not automagically make responsive content.
There was never meant to be a mobile silo. It's a mistake to think that 'Mobile First' means you go and design your website or web app for mobile first - as if in some kind silo that says on the door: "we're doing mobile in here". Mobile First is very much about removing those silos, not creating one for mobile.
Mobile First is a methodology – a mindset. It’s about designing with the constraints and capabilities that mobile provides. Designing with constraints forces you to focus and prioritise while designing with new capabilities attracts innovation. Hopefully with the result that what you learn from the process benefits your content everywhere - not just on mobile.
While 'Mobile First' is relevant, it's not the whole solution - not by a long shot.
Personally I think the word Mobile in 'Mobile First' has been taken a bit too literally and maybe with a bit too much tunnel vision - as a command rather than a mindset.
So don't take 'Mobile First' as your only design strategy - don't focus too much on mobile, but don't ignore it either.
Perhaps a better term for the approach might be "Content Everywhere".
This is kind of backed up by Joe Stewart whose design company Work & Co redesigned the Virgin America website:
In terms of the overall way to think about design and design process
or responsive, some people like to say there’s a mobile-first way of
looking at things but with responsive it’s everything first. It
actually is sort of relieving to be able to think about all of it all
To respond to the edited comments in the question:
It's worth trying to find case studies of how others have approached mobile for complex lines of business - such as the article or slideshare of Neil Turners experience working with TUI.
Below are a few relevant slides from my own slideshare on mobile UX
The fact is - mobile is here to stay. In fact, let's forget the word mobile for a moment - content can be consumed on a multitude of devices. Somewhere you feel you want to draw a line that says big screen experience on one side and small screen experience on the other. That line doesn't have to be your decision to have a m.whatever.com site.
More and more, the question is not about delivering different content for a big or a small screen, it's about how to make a consistent experience when moving between devices.
Having a long term content strategy that includes small screens is critical. Having a different strategy for mobile vs desktop is a road to disaster.
Think not about delivering different content, but about delivering content differently.