My school did something that seems not unlike what you're describing.
First, you went to each sorority for 10 minutes. This was followed by open houses, where that evening and the next day, you could go in and out of each sorority's rush room as you pleased. Then sororities would make their first round of cuts, generally cutting anyone who didn't make grades or who didn't stop by during open houses.
Next was "informal rush" - a few hours in the evening and the next day, where you could visit any sorority that had invited you back, pretty much for as long as you pleased (there were caps on time but they were pretty generous). You then attended a theme party with each sorority that had invited you. Sororities would then invite whoever they wanted to pref, which was the following evening.
From the PNM side, you could very easily mess things up. Each sorority only had 10 minutes to make a first impression, so you had
one convo, and if you happened to be paired with someone you didn't click with - you might decide not to go back during open houses, and you might miss out on what would have been a good match. You also had to be proactive if you wanted to get to all the sororities during open houses - you had to stand up and say "I want to check out my other options" and worry that you came across as rude or whatever. (It was forbidden for a sorority to prevent you from leaving.) And sometimes PNMs would spend the entire open house period with one sorority, essentially "suiciding" right at the start of rush, then if that sorority didn't invite them back, they were SOL.
From the sorority side, I was a member of the smallest (by far) sorority at my school. When PNMs came through tours, we had to use the time to do a skit rather than convos, because it would have been about 3 PNMs to each sister. PNMs saw the small number of sisters, got worried, and ran scared. It might have been beneficial if the PNMs were "forced" to give us a second chance - often, those who
did come back to open houses, and who were otherwise good fits, wound up joining - we were listed first on a lot of pref cards.
