I cleared the path near me recently for the same reasons and normally have hedge trimmers in my bag.
But... the reason it should the "council's job" is that they have fancy sit on lawnmowers which can do what took me two days in minutes I would expect. And surely that is the whole point of a council, we all club together so they can buy proper equipment for this kind of thing and cut a thousand paths with it rather than just the handful I use. I cut my own grass but I don't expect everyone to got into the local park and cut a square metre each.
Some councils clearly do get it, there is a cycle path I use which is well lit, swept a few times a week, drains cleared and everything cut back.
However, head in the other direction and part of my nearest NCR goes through a long, unlit tunnel with a surface borrowed from the NASA moon landing set. Apparently there is "no budget" for lighting although the road it goes under it lit up like a Christmas tree.
If your local council doesn't prioritise cycle paths it is important to badger your local councillor because they can do it properly if they find a fraction of they money they spend improving things for cars.