The thread title is perhaps a little 'limiting'-
Third rail is old fashioned and should be done away with
I can't see that 'fashion' should be an issue; nor that something (as with someone) should be 'done away with'- tho the language does draw the eye

So what is the issue?
Is third rail unsafe, or expensive to maintain, or unsatisfactory in some other regard? AIUI, the ORR does not support its use where it is currently not in place, citing safety concerns, principally for trackworkers and trespassers. That is a logical policy.
To 'do away with' existing third rail provision (and associated infrastructure, rolling stock etc) will incur costs and inconveniences- analysis and decisions will need to be made as to costs and benefits.
Technology provides possible ways ahead, and 'climate change' also provides impetus, eg to decarbonisation. So extension of, and in this case conversion to 'standard' 25kV OH may makes sense. Electro-diesels and Bi-modes also show that not every metre of every track needs to be electrified.
Could Clapham Junction be travelled through without wires? Could a train arrive and depart Waterloo or Victoria, or Weymouth or Brighton with a 'quick-charge' in between?
Could the third rail be allowed to 'wither on the vine'?