Background info: I am ARAM only player with around 3K games behind me with 70+ champions owned. I play 2-6 matches per weekday, often more during the weekends. The below is just my personal stance on the matter, but it might be close to the view of some other people out there. Oh, and I am quite long winded person, so I have added a tl;dr at the end.
I believe that having ARAM be "All Champions" would reduce the number of people fighting in Howling Abyss. I might still continue maybe, even though I would not be too happy of getting even more champions to suck with. I have plenty of those under my owned champions already (for example almost all the assassins I own). I really don't know what kind of pleasure I could get from a game with almost 60 more champions in my pool - I am not that much of a masochist.
If ARAM would turn into "All Champions" I might decide to take it as a challenge. Since if I suck with some champion, I usually try to figure out a way to play him differently and sometimes it works out nicely like in Evelynn's case (and with that success I even bought Shaco), often it doesn't and it is back to drawing board time . However, since I only play ARAMs, I would have very few chances to try the "New Improved Champion X" build, especially if the pool had all the 130 or so champions to roll from. In my case it would end with almost 50% of games playing with pretty much unplayed champion. That might kill the joy of the "challenge" for me. (For example the Shaco I mentioned above: I have had him around 6 months and I have managed to get him 6 times. If the pool was "all champions", I would still be at 3-4 tries and thus I would have made very little advancement in my [still newbie level] Shaco skills.)
And now to the reason why we are even talking about the matter: ARAM only accounts
I don't think ARAM only accounts are really that numerous as people think they are (the definition of ARAM only account is quite flexible though). It is just that most people have the strong ARAM champions and most sane people won't reroll them. Bad champions get rolled away and have higher chance of becoming strong ARAM champions than not as a whole. It is like a game of yatzee, you keep to good dice and reroll the bad ones hoping for good result. As the result the team becomes stronger, giving illusion of having "ARAM only accounts". (Yes, they exist, but aren't that common I think)
tl;dr: Thus I prefer the current "Champions Owned + free champions of the week" pool to the "All champions" pool. This way I can slowly increase my champion pool and increase my skills with them in more controlled fashion. ARAM only accounts are a fact, but much fewer in number than it appears.