It's an odd mixture of a document - it starts as pretty much as a marketer's pitch, but towards the end it does look at the disadvantages. And then there are things that are missing.
The biggest omission is anything serious about how to increase capacity. Overground was an easy gain, in that its use had declined to a low level, so when passengers did want to return to using it the capacity was there. In this quote:
The Mayor and TfL^s priority since 2007 has been to extend the London Overground network. This has happened in part through the addition of new track infrastructure, and more significantly through the devolution of suburban services on the Greater Anglia franchise in 2015.
what fraction of the increase was due to new track?
Closer inspection suggests that what they are really proposing is higher off-peak frequencies, possible because the longer-distance services are then less than in the peaks. That would mean more trains off-peak than in the peaks - as we once had for Waterloo-Reading. All very useful, but hard to pay for. In that context, this quote is interesting:
The recent extension of the London Overground following the devolution of suburban services from the Greater Anglia franchise in May 2015 demonstrates the challenge facing TfL. Prior to devolution, the previous operator, Abellio, ran these routes profitably. However, figures provided by TfL indicate that the service will now be run at a loss for the foreseeable future: between 2014/15 and 2020/21, TfL^s net expenditure in just these routes will total ^78 million. This is after the ongoing grant from government to TfL has been taken into account.
In terms of "which services do they want to grab", the map (fig.7, p 20) is a bit fuzzy, but does suggest nothing off
GWR▸ other than what becomes Crossrail (which is logical) and none of Chiltern or the other north-bound lines (which looks a bit less so).
SWT▸ would split at Virginia Water, with Windsor and Weybridge services being devolved.