I won't lie, the moon cycle with its leap month, complete (full 30) and rejected months (29 day) and complications around the end of month observations ... the moon is a challenge to incorporate into our timekeeping. Although struck by the elegance of certain solar calendars, I have to make a stand that I really believe the original Hebrew calendar was a lunar solar calendar and not purely solar.
These solar calendars often look more elegant on paper than in real life where questions would need to be asked about real or mean averaged equinoxes and solstices and whether these should be understood as from Jerusalem or from the equator. Jerusalem would be a problem to the perfect 91 day illusion of progression between these points. So much that could be said.
Sadly, English translations have not helped, because month no longer has its original meaning and the Hebrew "yerach" would be better translated as lunation or moon cycle than month. I hope this teaching will encourage you to do months according to the moon whether Hillel II, Conjunction, First Sliver or whatever other version of a lunar month there may be. The moon is a challenge, but it is beautiful and I think teaches us the deep lesson we need to learn about control and predictability.
Many blessings