On the topic of analog controls, I see absolutely no reason why classic shmups wouldn't benefit from this, it is a genre where more exact controls would only improve the experience, and I am really puzzled why it is still only a minority of the games using this. However, personally I prefer digital controls, I don't know why, but I guess it has something to do with tradition.
Also, two personal notes on my thoughts about what could make shooters more accessible in the future:
1. Competitive play.
I'm relatively new in shmups, in the sense that I probably didn't buy my first game in the genre until around 10-12 years ago. I used to hate games like Contra and 1942, which were generally considered mainstream games back then (and I'm a HUGE Contra fan now!), and one of the most popular genres too - basically my view on shmups was pretty much the same as on racing and sports games, and first person shooters when those came around. I wanted fun and easily accessible games like platformers and adventure games. As a kid I even returned Guardian Legend to the store when I found out it was a shoot'em up (found it in the bargain bin, and back then all you had to go by was the images on the back of the package

)
It is only much later that I realised how games like this are pretty much the core essence of true action based gameplay and started embracing them.
The interesting thing about this otherwise completely pointless anecdote is that out of the genres that ruled supreme back then, the shoot'em up is the only one that pretty much died in the mainstream point of view. People view it as a "retro" genre, and even though they might find the game fun, they still see it as old school, and not worth paying full price for. Sports and racing games have constantly evolved since then, and even though they are all so alike that every new game in a serious pretty much replaces the former, they are still among the very best selling games, and people will gladly pay full price for the latest FIFA game -
every single year-!
In other words, it is not the old school arcade style approach of the shoot'em up genre that makes people consider them dated, and I don't think it's the 2D gameplay either (essentially a football game is semi-2D as well), I think there are two major factors keeping them apart.
One is the fact that the games that remain popular are simulation games. For some reason people want realistic games, and sports and cars are generally considered "cool", while dragons and space ships are geeky. I hope a game like Bulletstorm (a concept which I otherwise hate) will help change people's view a bit here, but otherwise shoot'em ups are pretty much doomed here.
The other fact is the competitive gameplay which keeps people coming back to these games! People make tournaments, visit eachother to compete, and play online constantly. If shoot'em ups had a more direct competitive gameplay, it would make them MUCH more interesting to the general public.
Online leaderboards (Xbox Live) has already helped a lot in brining back people's interest in high scores (Microsoft does deserve a lot of credit for this), but it's not a very fun way to compete for most people, as you don't compete head on with your friends, and the best player will always be in top, where games like racing games and fighters always give both players a chance of winning if the opponent makes a mistake.
I don't know how it would work, but if someone could make it work, it would be grand.
2. Credit feeding.
Frederik sort of covered this in a post further up, but I'd like to accentuate on its importance.
I don't think the challenge of the typical shmup really puts people off. Recently the way every mainstream game holds your hand and basically makes it impossible to fail has been getting a lot of criticism from the "hardcore crowd" (ie. everybody who actually want to play a game as opposed to watch an interactive movie), but I think the idea that a game has to be easy to overcome in order to sell is completely fake.
Take a couple of game that sell really well:
Pretty much every racing game is really tough! You need to be an expert to beat the harder stages, you need to perfect every curve to get a serious chance of winning, and the only way I could beat GT5 (that's right, I played a racing game, I'm pretty shocked myself) was by spending all my ingame credits on tuning my cars, giving me a completely unfair handicap.
FPS games! The plague of the mainstream, a billion similar world war 2 themed games in a genre that has remained largely uninspired for more than ten years (sound familiar?

). Most people who are really into this genre don't play it for the single player campaign. They play it for the online matches, against real people, whose skill probably match their own, giving an average chance of winning of merely 50% even for a very skilled player, playing against a similarly skilled player! People play it for the challenge, so why shouldn't they be able to play shoot'em ups for the challenge?
Anyway, getting back to the point, I don't think the extreme difficulty of many shooters put people off (but the first few stages SHOULD be possible for novices to complete with some training!), I think the fact that you can always pop in another free credit and play until the end of the game, makes the game appear too EASY to a person who doesn't care about score attacks.
Another terrible effect this has is when you play the game with infinite credits, you automatically take the game less seriously, and usually fail a lot more. You never get the training needed to improve your skills, creating even worse results, up until the point where you spend 20 credits on the end boss alone. This isn't fun for anyone playing the game, but a lot of people actually end up doing this if they get free credits!!
I think games like Ikaruga, Contra Shattered Soldier and Gradius V found the perfect way to do this.
Basically give the player a low number, eg. 3 credits to play with. Most people won't be able to complete the game with this few credits, and IF they are, it will be fine for them anyway. This will give newcomers a good challenge they can't cheat themselves through, and by rewarding the player with additional credits the more they play the game, they slowly get a better chance of completing the game, that will still pose a challenge even though they don't care how many credits they pop.
The trick here is that not only will these games be able to challenge both novices and experts, it will also allow novices to grow better at the game as they play, basically FORCING them to effectively train their skills until they are able to enjoy the game as much as an expert of the genre, and move on to score attacks!
The games I mentioned (with the possible exclusion of Ikaruga) are all mainly survival based games which means this approach makes more sense, but I think it would also make a lot of sense in a scoring based game, like a Cave shooter. It wouldn't hurt these games to have an added "survival challenge", most of us never play these games on more than one credit anyway!
tl;dr? Sorry.
