I am not convinced need to run to main works was actually a design requirement.
I am more of the view they did it because they could; not specifically designed but a built in nice to have rather than a need.
It was Lancing works, not Eastleigh, that was the traditional 1940 stock works, with other work done mostly at Stewarts Lane. Where they were seen at Wimbledon etc under their own power was usually test trains inconjunction with those movements. See the item I cite below, they had visits and other test and trial trips to e.g. Brighton and (1970s) Farnham.
After Lancing closed, their regular works became Eastleigh, but I am not convinced they did that move under their own power, or, at least, not all they way, as west of Pirbright Jn was [then new] 750 V not traditoonal 630 V. Farnham, Brighton, etc was 630 V hence re. above allowed. Took a few more years to get to 750 V everywhere, legacy stock had to be rewired first.
Even Subs were limited at first to Eastleigh. As routine overhauls continued, a number of Subs, only those with EE507 motors, were rewired to 750 V and those were permitted to run under their own power. But non rewired Sub en route to Eastleigh works, and from works if not rewired (and or retained 339 motors) were either loco hauled or tractored by other EMU.
See
Should have all anyone ever needs to know; although does not mention the 750 V issue; I will take that up privately with the author over a beer when next we meet.
Possbly the mists of time have muddled W&C tube stock main line runs with IOW stock, which was 750 V as part of the work to convert them from LT to BR. Another good read to demolish a number of myths and legends is here :