I've seen people use plastic cable-ties to hold down the schach in their sukkah (as they're not "makabel tuma"), but others who don't like it, I think because they're not plant matter? Can anyone clarify who holds what, and why?
There are two issues:
The first is the more commonly accepted issue. There is a question as to whether you are allowed to hold up your schach with something that is pasul for schach. The common example was to use material with a bais kibul, so it is referred to as "maamid b'davar hamekabel tumah". The issue, according to those who assur, is that you may come to use these materials for the schach itself. An exception to this rule is to hold the schach up with something that people would not use for schach (shading). For this reason it is mutar acc. to everyone to rest the schach on a stone wall. Plastic is something that people use as shade and would be part of this issue.
I cannot say that the minhag haolam is to be machmir since I have seen many who rest their schach on metal and plastic poles or sheets, but bnei yeshiva are machmir and Rabbonim are machmir for those that ask, unless, like Dave said, this is your only option for holding the schach down. You also needn't turn down an invitation for a meal for this issue as halachically it is considered a chumra (maase rav).
The second issue is from the Chazon Ish (I did not see this inside, but was told by a well known posek who told me that he is noheg like the Chazon Ish), which is not to build the entire succah with a davar hamekabel tumah because of Tahara. This is not an issue for your plastic ties. Nor have I heard of any one else machmir for this except the aforementioned.
Piskei Teshuvos (Vol 6 pg. 328) cites Avnei Yoshfei #120 who says that plastic strings may not be used since they are not "gidulei karka," as you mentioned. He notes, however, that this would only be a problem if the schach would otherwise blow off in normal winds.