But the existing DC distribution system design assumes that it is all connected in parallel, and DC regeneration also requires the system to be connected in parallel to improve receptivity. If the third rail itself wasn't to be live, you'd need a separate high current path running alongside it to provide the same traction current if it couldn't be passed along the third rail...
I'd assumed that to be part of the idea anyway. Third rails at present are laid in relatively short stretches, with a gap of a few metres to the next stretch spanned by a pair of heavy jumper cables. So to my mind the immediately obvious way to do it is (1) instead of chaining those short sections to each other, supply them from a heavy bus bar alongside the track; and (2) run each set of cables from a rail section to a bus bar through a switch. So everything still looks the same in terms of the "all in parallel" assumptions,
Since the bus bar does not have the dimensional and mechanical constraints of a third rail, it can be eg. a big chunky bar of aluminium to improve conductivity, which would make the system less lossy. Also since no more than a couple of hundred metres at a time of the third rail itself would be contributing to losses, it might be possible to use a smaller section that would be cheaper than the present standard.
This has been proposed several times before on other threads and the answer is that the switchgear would get hammered switching on and off so frequently, as well as the further potential for delay if a fault develops. Many parts of the southern region have trains 2 or 3 minutes apart for substantial portions of each hour. Trains are up to 240m long with very close signal spacing, the length of sections and continuous switching in and out would hammer the system.
Not if you use solid state switchgear, and these days I'd not imagine using anything else. Reliability is now pretty much bomb-proof. The control system need be no more than a unit associated with each switch that detects if the load side is connected to a train, which is not difficult. As an emergency fault bodge until someone can come and plug a new switch module in, you could simply make a temporary bridge to the adjacent section of third rail with jump leads.
AC conductors primarily conduct along the edges so no need for the mass of 3rd rail.
Actually that is a nuisance, not an advantage. To handle the same current you still need the same
effective cross sectional area of conductor. But when skin effect is significant, the central part of the cross sectional area is no longer "effective", so it's wasted mass. So to get an efficient conductor you have to make it in the form of a hollow tube, which is more awkward than a solid bar.
The size of conductor wire used for overhead AC electrification is roughly one "skin depth" in radius, so it's not yet fat enough to have to worrry about that.