Paths to me are permanent structures since using them creates some compaction.
To counter this inevitable event, mulches of
wood chips are one of the best, fungi will inhabit the woodchips and work their way down into the soil beneath, giving the start of a great mycelium network that will only benefit the plants in the gardens.
Some people use gravel or small river rocks, this is fine if you are going to be using heavy equipment most of the time but not so good if you are just walking with a wheel barrow.
Weeds can be left but then you are also adding to the weed seed load of the area, most people really don't want to add to their worries or weeding time load.
However, if you are ok with using weeds for mulch around your plants, then letting weeds cover the paths so you can chop and drop where you want new mulch, then your paths are going to be fine with weeds for the cover.
I personally have a mix of weedy grass lawn paths (between raised beds) and wood chip mulch paths (less frequently used paths).
Most of our
raised bed gardens already have a good fungal network in the soil and the wood chip paths are spreading more fungal network to those areas slated for development at a later date. (these are paths to areas I am
cutting firewood, clearing for silvopasture and opening to the sun for more garden spaces).
We do not want any soil to lay naked to the sun or sky, all soil
should have something covering it, preferably growing plants but if foot traffic is going to be heavy then something more durable is usually better and that is where wood chips shine.
Redhawk