The only gated adits in the Via Gellia are Goodluck, Merry Tom to which access is arrangable, then there's Owlet Hole and Ball Eye to which no access is permitted. That leaves you with Houghton Pipe, Cowhole Pipe, Groaning Tor, Jacob's Dream, Old Gells, Silver Eye, Slaley Sough, Dunsley Springs, Henstocks, Thumper Sitch, and Spinney level, amongst others. Most shafts in the Peak have concrete sleepers on them, which with the right knowledge and equipment can be opened for access in about ten minutes. There are mines in the peak that aren't generally accessible but for the most part you can either just go in or get permission by asking.
As for health and safety gone mad, nothing of the sort. It's been established in law since the 19th century that a land owner or occupier has a duty to prevent the casual passerby coming to grief in an abandoned mine, but all this needs is something to prevent said passerby just walking or falling into the mine. Usually a fence is sufficient, or in areas where there are lots of small and not fully mapped shafts, warning signs at the entrances to the land telling people to stick to the footpaths!
Follow the horses, Johnny my laddie, follow the horses canny lad-oh!