Replying to the thread title, Tfl should absolutely not take over the branches.
There is a strong argument that they are suited to a different form of rolling stock (e.g tram-train, Class 230), allowing most of those remaining Turbos to head out to the West Country and displace some 150s to the tin can factory, but the short platforms and conflicting moves on all the branch line main line junctions would make any sort of improved through service hell to plan if they were transferred to Liz Line.
You could transfer the operations to TfL if you wanted, for a very slim discernable benefit in marketing, but the best thing to do is to order battery 230s if the Greenford line trial is successful, install fast chargers at both termini. It could even save a unit on one of the more frequent branches compared to /increase service frequency ,due to the faster turnaround. These could also be used on the Brentford branch line planned for reopening with minimal effort given another forum user mentioning a proximate Vivarail depot in Ealing and the limited infrastructure works needed to reopen (e.g. fast chargers and maybe a small on platform stabling shed at each end.)
This would obviously change if the Windsor link ever goes ahead (the tunnel proposal linking the 2 stations, presumably they would then transfer to Waterloo)/Bourne End to High Wycombe is reopened (extend the XR Maidenhead terminators though to HW, shuttle for Marlow from Bourne End although platform extensions may be difficult there) and Greenford/potential Brentford branch electrification (transfer to LO), but you reassess services at that point.
The discussion about extending XR to Didcot is a little absurd, those stations beyond Reading towards Oxford do not warrant a metro style service. When the Oxford line is redeveloped and electrified, 387 service can be extended to Oxford and then we can reassess the semi fast patterns. Presumably, at that point, the 5 car IET can be released to bolster services on non/partially-electrifed lines and 2 387s can be used to increase capacity on the semi fast.
But that's long in the future.