Most lines are having automated operation rolled out to them gradually as well; they just retain a driver to keep everyone happy. The busiest lines already are: Victoria, Jubilee, Central and Northern. The latter two were retrofits.
Notably, the Victoria line has no such doors, and currently runs more trains per hour than the Jubilee line, although this is slated to change, though it won't lead by much after the upgrade.
The Northern and Central lines were automated within the last few years. The Victoria was automated since it first opened in 1968. Once the sub-surface signaling upgrade is completed, the Circle, District, Hammersmith and City, and Metropolitan lines will also be automated.
Automated in this sense means "driver controls doors and presses button to send train to the next station". When the rolling stock is next refreshed, the cab might be axed and instead it'll follow a similar model to the DLR today - a single person roaming the train checking fares etc and issuing the "close doors/move to next station" command from a panel near each set of doors.
(1) Unions. (The tube driver's union is able to stop London and they are not afraid to do so.)
(2) It's not actually very easy to do for several of London's tube stations. Several of the tube stations have curved platforms making it rather harder to install the safety double elevator doors.
(2) Skip automation on these stations.
All lines will go to CBTC soon. Not having a driver doesn't add any capacity. Indeed it can lower it quickly in the event of something going wrong (may take a while to get staff to the train to fix it)
This allows Singapore to run a lot more trains. I don't think I ever waited for more than five minutes. Automation is possible in London. The DLR and Jubilee lines are automated. There's push back from unions; jobs lost and such -- but overall a fully automated tube system would reduce congestion by quite a bit simply due to increasing the volume and frequency in trains safely.