Many, many gedolim had children who didn’t follow their derech. The fact that most of rav moshes children were not only frum but gedolei torah who grew up in a hostile environment is astounding.
That he had one son in law who turned out to be modern is not surprising; most people at that time were not what we would call yeshivishe, either his daughter wouldn’t fit with a yeshivishe guy and she needs to get married to, or tendler was a bochur who didn’t wear his hashkofos on his sleeve…or it could be he hadn’t developed those shitos yet. The possibilities are numerous, but none of them point to rav moshe being accepting of zionism, modern orthodoxy, or violations of halacha including ascending har habayis.
There’s also no way rav moshe would allow breaking minhag makom, as the rabbonim in yerushalayim had long since forbade the practice, even if (and there isn’t) there was a legitimate heter.