Unforunately councils have ABSOLUTE power when it comes to shutting down an airstrip! (and they usually know very little about aviation) I know of half a dozen private strips belonging to flying buddies, most of them with no issues whatsoever because the owner spoke to the neighbours and they don't care. None of said strips bar 1 have council permits. A box of shortbread or bottle of plonk at Christmas or the occasional joy-flight for the neighbours go a long way to keep things running smoothly. I used to do this when operating from a 100 acre property and when flying low over houses on the approach.
One strip that I know of wasn't shut down by council however was limited by council due to neighbours complaints to one single aircraft coming and going only. Not a bad compromise, so no visitors allowed. Probably a blessing in disguise from a liability perspective.
Flying from your own property can be awesome and so very convenient, but you really need to consider what you are flying into it, like a Skyfox Gazelle at 60db versus a C206 at 120db...