If the Underground was built today how would it be different

I've never heard of the Adkins-Dawson system.
The Adkins-Lewis system is a helical shaft drive mounted centrally between the running rails.
There is no 'grip'. The shaft turns continuously, and the speed of the cars is governed by the pitch of the helix. There are no 'stops'.

Bit of a future-gaze, but personally I'm quite interested in whether there's a place for Personal Rapid Transit in the metro space. I'm imagining a very small train 'capsule' with 4-8 tip-up seats which are fully autonomous (routed by a central computer), and can provide a one seat ride between any pair of stations systemwide.

During rush hour, passengers can be grouped together using intelligent software, e.g. tell the system where you're going as you enter the station and it'll group you with a couple of others who want the same route. And perhaps the cars can run in groups.

I realise you'd lose space efficiency from the smaller sized 'trains' but you'd gain a lot of flexibility being able to run the 'cars' very tightly together and you'd have no interchange passengers at stations which would save capacity there too.

Hah, for sure, it's not gonna work on a line like that! 

(back of the envelope calculations when I should be out in the garden)

I think it's a fairly simple calculation of floor-space and time. For a given available width, you can sit X people across; call it 10' and 6 people. A small carriage will need seating plus front/rear buffer space (4' each?). So a 12-person carriage will need two rows (5' each?) plus maybe 8' buffer equals 18' for 12 people. Hardly efficient, space-wise.

If you could make those work, there's still the issue of headways and switching. Coupled up into trains, this could approach current TPH headways (although probably not capacity), but individually you'd still have to time-separate the carriages at least somewhat (5-10 seconds?), which further reduces the track capacity. Add "dead time" around facing points movements and that further reduces throughput. Oh, and accel/decel times as needed.

I'm sure there are other factors I've missed.

I think it's notable that in 200 years of "gadgetbahn" "innovation", nothing has yet to properly supplant the basic traditional railway in the roles in fulfils and has fulfilled for the past 200 years.

On the flip side, the canals were a dominant transport technology for about 100 years. Railways came along, were better, and so more-or-less replaced them. The fact railways haven't gone the way of canals tells me that they're still just about as good as doing their job (if not better) as they always have been. Thus there's very little need for concepts like PRT, maglevs etc. in many, if not most circumstances. They have their place, but that place is almost always a very very small niche.

There were many occasions during the early introduction of D Stock that trains were “tipped out”, “detrained” at Bow Road westbound due to the doors failing to close or even budge when fully loaded.
It was thought that the cars left Birmingham in with a slight upward curve to the under-frame and with a heavy complement of passengers this straightened out the under-frame but then buckled the door apertures.
After adjusting the door runners, the problem gradually disappeared as the cars aged.