An overt rejection would not use months in any form. The lengths of months don't "make sense", but they're not derived using strict mathematics for what would be most logical; they developed bit by bit, with Roman authorities trying to correct things when the existing system proved to be incorrect (what we have now, from Julius Caesar's reforms, is far simpler that what they had before, which had entire months (thought not quite as long as other months) inserted every so often to keep the months and years roughly in line - see https://en.wikipedia.org/wiki/Roman_calendar and "intercalation" ).
I don't think our calendar does "alienate us from the natural world". The Julian, and then Gregorian, reforms were done to keep the day of the year for which a season starts as constant as possible - that's specifically keeping us in sync with the natural world. The length of the Moon's period (and thus the pattern of tides) just doesn't fit easily with either the length of the day or the length of the year, and that is, quite reasonably, the one that is not kept strictly to (days and years always have huge effects on everyone, the brightness of the moon or the height of the tides, less so). The natural world doesn't divide up neatly.
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