If they give you a specific deadline, then go with that.
It gets hairy when we extend the invitations (especially if it's a specific date or sets of dates) and then we have candidates not responding, asking to switch, and/or have others to invite, but have to wait to see what's open. Services like Interview Broker are helpful in these instances, but the flip side is that we can't offer a personalized panel of interviewers doing it that way. Some programs attempt to pull together interviewers based on candidates' stated sub-sub interests if at all possible.
In general, candidates' personal statements are more closely considered after the apps clear the filters. Of course we look at things like whether candidates can express themselves well and don't have typos, but it's also an intro to your personalities, interests, and ambitions. Will it make or break whether an invitation will be extended? Rarely. How it can kill the possibility of an invitation is if it's clear that a candidate doesn't have a strong grasp of how to write in English.