I think OP has a reasonable request. Missions that players take are specific to them, it doesn't affect anyone else's ability to get missions (unless it's a wing mission and you share it obviously).
People talking about living universe and whatnot as a response are being a little silly. Regardless of whether you complete a mission or not, anyone else opening the board sees pretty much the exact same types of missions. Over, and over, and over. Most of these responses against the OP's suggestion are not really relevant imo.
The only exception would be missions that are specific to the faction or system States, which will obviously stop generating once those states end. Even now, you can still complete those if you have them once the state ends, so what the OP requests wouldn't change anything for those either. Arguing BGS is also rather silly; the OP is still going to do missions either way, the impact on the BGS remains. He just wants the game to not punish him for having less time to play so he can maybe finish the occasional one that he didn't get time for after accepting.
Now, I personally think the mission timers are quite generous anyway (too generous in many cases...imo), but I can see how someone who doesn't have time to play every day could be unhappy that a mission they thought they had time to finish today but didn't would not appreciate that it then times out before they can log back in.