The explanation is as follows:
PAA’a quote of the Magen Avraham is obviously very relevant, but missing in the reasoning.
According to kabbalistic sources and led by the Ari Hakadosh, the night hours before chatzot are the hours of “din” and not rachamim. The 13 middot should not be said then, to the point that some sources regard it as close to kefirah (mekatzetz b’netiyot is the term used). Not a single sefardi posek permits slichot before chatzot, and some of them prefer a person sit silently and say nothing if he is “stuck” in a shul where they are saying slichot before chatzot. Until Reb Moshe, there was no Ashkenazi posek who disagreed, as can be seen by the Magen Avraham.
In the old days , this was not an issue, as slichot was always said in the early morning, b’ashmoret haboker, during the entire slichot period, including the first night. Clearly, the piyut of b’motza’ei menucha still applies b’ashmoret haboker.
For reasons that are unclear, maybe due to the advent of chazanut (my opinion) , perhaps due to the difficulty of getting up so early in the morning, something that not everyone did, the first night became an understandably big deal (with outstanding attendance by all measures), and was moved to motzaei shabbat, but only after chatzot.
When it became dangerous for individuals to walk outside late at night in America, selichot gradually was shifted to before chatzot. The reasoning is simple. It is preferable to say it at night before chatzot than not to say it at all. Although this is against kabbalah, Reb Moshe as a Litivishe posek par excellence could find no issur in saying it before chatzot and he reluctantly permitted it as a ho’ro’at sh’ah. Others followed quickly, and saying selichot nowadays before chatzot is a common practice among the halachic-bent ashkenazim. I have never seen it amongst the sefardim, certainly not here in Israel.
A last point. 147 mentioned the fact that we say 13 midot on yom kippur night, in contradiction to what I explained above. Although this is a good question, my opinion is that the “din” issue of those hours falls away on Yom Kippur, in which the sha’arei shamayim are open “m’erev ad erev” through ne’ilah, lechol ha’de’ot.