I always had all of my embryos make it to Day 5 and I had a lot of embryos go to day 5. The first time, they froze 1. The second...two and the third time..none. At my clinic they would only freeze embryos that were fully expanded by Day 6 and were graded at least a grade 2 on Day 6 on a scale of 1-3. In their opinion, if an embie is not fully expanded by Day 6, it is an abnormal embryo. So..no..my clinic would not have waited until Day 7 to see what happens. If an embie was not fully expanded by Day 6, they would discard it.
IMHO, some clinics, especially those that do not vitrify, will not freeze embryos that otherwise should be frozen because they know that only the best will likely survive the thaw under the old freezing methods and they do not want to ruin their "thaw stats." On the other hand, you want a clinic with standards. For instance, CNY used to freeze EVERYTHING....good or bad, giving IMHO people false hope. They no longer do that thankfully.
It sounds like your clinic has standards and to me, that is a good thing. You would not want to waste your time, money and energy on FET cycles that have no chance of working out.