I suppose with games the other factor is that people tend to rent them far longer than they do a film. Which in turn means there is less in the rental pool to be sent out, which is likely why people aren't seeing the games they want.
With LoveFilm a couple of years back, they gave me an extra rental, so I got Assassins Creed 2. I ended up having the game for about 6 months. I also rented some Wii and Nintendo DS games which I took with me when I went to France for 4 months.
Perhaps with games, services needed to introduce a time limit on them, which obviously moves back to the old high-street rental model.
I suppose there may be some truth in games going missing more. I'm sure I remember reading somewhere that LoveFilm has a kind of three strike rule on discs reported missing before they cancel your account. I for example have been with them 6 years and in that time, only one disc has gone missing in the post. So if somebody signs up, rents games and they all seem to disappear, it makes sense to cancel their account, but probably difficult to prove its stolen and charge them.
I think though that there is also a lot more to it, in particular what Publishers want in fees for being allowed to rent their games. Its quite well known how displeased they are with the preowned market and likely rentals aren't much different. If they are asking a high price, it doesn't work so well for the business when they are trying to keep their rental fees low, yet people will have the games for a long period of time. They don't really want to buy more of them when really they need to be turning the discs around quickly.