Yeah, it's hard to visualize. It took me a long time to see it when I first chose to mount a parting tool back there. You're right that the cutting forces at the tip of the tool are directed upwards but it is the tool post that changes things. As I see it, the locking bolt that holds the tool post down acts as a fulcrum and changes the direction of the forces so that they are downward at the rear of the cross slide. Any clearances under the cross slide are taken up and the cutting forces are then transferred to the structure of the lathe. My very rudimentary readings with a tenths indicator suggest that all of this is happening.
There is no question that rigidity is enhanced by rear mounting a cutting tool, whether it is a parting tool, knurling tool or turning tool. How it happens is not settled (except for me) but that isn't what matters. What matters is that it works. If you have not rear mounted your parting tool back there then you really should.
Insofar as the query from the OP, the lathe turns well from the front. How well depends on the tool geometry.