Bedford is 80 km away from London - normally where outer commuter trains end, including Milton Keynes, Basingstoke, etc.
Anywhere further becomes a regional express type operation, for example, the distance from the London Terminal to Corby is similar to the distance to Portsmouth / Southampton.
Therefore it is inappropriate for Thameslink to run anywhere further. In fact I don't think Peterborough trains should run into the core as well - the problem on this route is the 80 km limit lies on nowhere major and the next major destination Peterborough is very far away.
That would mean people living in Brighton would loose out on services other than the core Brighton - Bedford services, that go through London and provide in many cases vital connections to other routes where only one change needs to be made, whereas you take out the trains to Peterborough & Cambridge, then two or more changes need to be made for the same journey.
Brighton has recently had the GWR services being stopped, pretty much due to the lack of stock that GWR has in it's fleet and also lack of funding from the Government, so you would be penalising those of us that live on the South East coast if you took further services away.
What would be better though, since there is stopping services from King's Cross to Peterborough, is to make the services from Brighton semi - fast on the route if timing can permit.
With regards the EMR Connect services to Corby from St Pancras, it would be a disaster to add those to Thameslink due to the many points been made by others in this thread.