I put myself in the 'younger generation' and from my perspective, young people definitely want to shoot but from what I have seen, most don't even know that they can.
I keep my shooting to myself, as in I don't broadcast it online and don't discuss it with people I don't know. My friends and family that do know frequently ask me if they can try it, especially those around 30. In the last 2 years I have successfully gotten 5 friends and colleagues into shooting who have joined clubs and got their FAC.
All of them said the same thing on their first try - "I didn't know I could do this?!?". Even when I was serving, it amazed me how many people had no idea what you could shoot as a civilian.
I've often thought that clubs and representative bodies don't advertise or make the wider public aware of shooting, maybe due to possible negative attention? Quite a lot of clubs I have seen recently are full and aren't taking new applications. How are young people meant to get into shooting if they don't know about it and in some cases, can't find a club to start at? Even guest days at Bisley are only attended by guests of (please correct me if I am wrong) NRA members who have invited them. Shooting as a sport doesn't seem to be outwardly/publicly facing and the community is introvert. As above, probably due to negative attention which it would undoubtedly receive if it pushed out a 'come try shooting, you might find it fun!' campaign. Imagine the newspaper articles...
The NRA push out some good coverage of what happens at Bisley over Facebook but I'm still yet to see an advert for a local range or organisation come up. You only see those pictures if you go searching for them and follow the NRA specifically. Again, perhaps less is better? It's a difficult one.
One other thing I've noticed is that a number of younger people come into shooting with an interest in modern rifles and disciplines, aka 'black rifles' and IPSC, PSG, Gallery, CSR etc. These attitudes/interests don't seem to be received with the same enthusiasm as they would have perhaps been received 25 - 35 years ago, pre-pistol and semi-auto bans. Maybe I'm wrong, it's before my time but seeing the lad down the end of the line getting remarks because he's popping away with his .22 AR-15 and the next guy with his straight pull AK (or MARS rifle

) suggests to me that there's a gap which is perhaps generational? Similar to music? Maybe I've just seen a few bad examples.
I started shooting properly at 13 years of age and the first club I joined only catered for prone shooting with an Anschutz, jacket and sling. I enjoyed it but we had young people turn up, shoot 3 or 4 times and then never return. The older pax would ask me "why?". The simple answer is (in my opinion) that they want to shoot more varied/exciting firearms and disciplines. I guess it's like wanting to get into a band at school, going to your music lesson and being told that you can only learn the recorder. It puts you off.
But yes, there is also unfortunately a large proportion of youngsters who see firearms as evil, but hasn't that always been the case?
I'm typing this from the US the eve before going out with an AR-10 to ping steel from 100 - 1000 yards and then burn through 400 rounds on an M60, so I can't complain.