I agree, to be honest, watching target sports on TV is for most people (not target shooting nerds like me) an experience akin to watching paint dry.
How do we improve the image? We can't really make 10m air rifle or pistol dramatic can we? Similarly most of the rimfire disciplines aren't exactly going to give viewers an adrenaline fix.
The clay shooting however has some movement that viewers who want action may appreciate, possibly with the right TV presentation moving target (running boar/deer targets) could give a non-shooting viewer a bit of excitement.
How does shooting become more acceptable to the general public? TV and press exposure will only come if we have a young pretty female star getting gold medals for one of the UK countries, then the mass media might be interested in putting a few pics on the front page. Without being sexist the general public would not be interested in seeing pics of blokes with guns, girls with guns would get a bit more attention but how long would the attention span of the nations paper buyers last? The general public seems to have a goldfish type attention span for anything but football.