You are correct; the London Thameslink ticket would not be accepted at Charing Cross. It is only valid as far as London Bridge on that route (or Blackfriars, if you prefer).
If you want to travel to Charing Cross, I would suggest entering that as the destination to ensure you're offered appropriate fares.
The way the Southern/GTR sites operate is as
@OscarH describes; the suggested fare is the cheapest one that's valid for the selected journey, even if the fare destination doesn't match what you searched for (if this wasn't done, people would often be overcharged as it's regularly cheaper to get a ticket that isn't to the exact location you have searched for).
You are given the option to select a more expensive fare towards the bottom of the screen, under the "See all available fares" button. You need to select the £15.85 fare to London Terminals, routed "Not valid for travel on Gatwick Express services", for a ticket that's valid to Charing Cross.
As it happens, it would be cheaper to walk from Blackfriars (you'd want to be at the very front of the train for the exit on the north side of the Thames). If you'd prefer not to walk, you could buy a London Bridge to Charing Cross Anytime Day Return for £4.75, making for a total of £14.85 - still cheaper than the above fare (albeit less flexible as you'd be tied to travelling via London Bridge).
Using the bus from London Bridge to Charing Cross would be cheaper than the train, but admittedly slower and less convenient.
