In general most work in any professional job will tend more towards small cog in a big machine kind of thing, whether in the UK or elsewhere like the US. If you're working on space travel or electric vehicles then you would probably be very focused on a single motor design, or even a component of that motor. The exceptions would be either later on in your career or getting into something which is at a very early stage of development. At some small companies you might do more but they also often specialise on more specific things or less fulfilling things as you put it.
If you want to do something prestigious, which I think is what you're really trying to say, then you need to have an extremely good eye for current and future trends. And that will often mean working from a desk.
If you want to be up and about a bit more then you will need to pick your job wisely, or you might be better off not going the degree route and going the apprenticeship route instead.
I'm currently doing my PhD on Wind Energy and specifically a fairly big picture topic within Wind Energy, but I didn't choose it because of those things and I don't know any other PhDs who are in a similar situation. Most PhDs are involved in a very specific topic which lends itself to small cog in a big machine again.
My coursemates went on to work at major companies like Airbus, JLR, McLaren, and also smaller impactful firms like Reaction Engines and Volocopter but their situation was pretty much the same.
Major engineering projects are just far too complicated nowadays for anyone to solely develop major components within them.
If you really like engineering then you'll still enjoy it though.