We should make the distinction here between PGS and PGD. In PGS (or CCS) there is debate whether ICSI is necessary. In PGD, for the moment, there is no debate: ICSI must be used.
We have a rich course on PGS (LINK) but here is the crux as it pertains to ICSI. PGS theoretically helps the doctor determine whether an embryo is chromosomally normal (or not) so the doctor can prioritize which embryo(s) to transfer. To get this information, an embryologist must cleave a few cells from the outer layer of an embryo for analysis.
Most patients who want PGS performed on their embryos are told this is only possible if those embryos were created using ICSI. There is debate on that point.
The argument for using ICSI when doing PGS is that it makes PGS more accurate. When eggs are fertilized during CI, they are surrounded in a petri dish with droplets that contain sperm that compete to fertilize the egg. As a result, sperm may get stuck to the outside of the egg and leave their genetic residue behind.
The fear is when the PGS biopsy is performed, residue on the outside of the egg (now an embryo) may get included and confuse the analysis. In theory, ICSI eliminates this risk because a single sperm is injected directly into the egg with no residue left on the egg / embryos outside.
Some doctors disagree and point to the lack of data behind the pro-PGS assertions. They believe PGS is accurate enough that it won’t be thrown off by potential residue of other sperm.
Just as with PGS, PGD requires a biopsy sample of the embryo to be taken and read. However this time the lab is looking to detect a specific genetic disease when there is good reason to believe one will exist.
The risks (and consequences) for wrongly diagnosing a problematic embryo as “normal” here are enormous. The field is in agreement that using ICSI to fertilize eggs slated to be PGD-tested is the more conservative, sensible approach. We don’t expect to ever see data disproving this.