Is your point that you would like to see an extra minute added to the time (via the snooze button), or, that you would like to see, essentially, an "only one minute left b/f DQ" warning, and that--with this--you'd be fine leaving it at its current length of time? I would not mind that second choice at all, with an auditory signal as well.
From my perspective, in 99% of cases; there is no reason to not move for more than 2 minutes AND also not write a 10 second note in chat, explaining the reason for the delay. Well, actually, I can think of lots of reasons (kid suddenly throws up on the rug, cat knocks over a vase, wife/husband sets dinner on fire accidentally), but for all of those (ie, situations where you need to run away from the computer and not write a short note), my thought is that it's fine to accept a DQ...part of the trade-off for having a wife/kid/pet/vase in my life! :-)
[The one glaring exception to all the above is when technical problems on the site or on your end make it impossible to make a move. Not sure of a solution for that issue--other than lengthening the time to make a move, and I'd hate that.]
Hey, maybe that could be an option we could adjust? There would be some minimum amount of time (say, 2 minutes, or whatever) that players would always have. But in "Options/Settings," players who like to move through games quickly could pick settings that would only match them with similar "<2 minutes" players. Others could pick a setting of 4 minutes, or 6 minutes, or even unlimited time, and there would be the option to filter potential opponents.