Synthesis of benzophenone
Your basic strategy to involve a Friedel-Crafts acylation is correct, but your route is a bit circuitous.  You could acylate benzene directly to produce benzaldehyde in one step using carbon monoxide and hydrochloric acid to generate formyl chloride in situ.



While it is "just another" Friedel-Crafts acylation, it has it's own name, the Gatterman-Koch reaction, after its discoverers.  Once in hand, you could oxidize benzaldehyde to benzoic acid.

An alternate approach would be to 1) brominate benzene to form bromobenzene, 2) form the Grignard reagent with magnesium and 3) react it with carbon dioxide to produce benzoic acid.

In either case, with benzoic acid in hand, you can procede as you've indicated, on to benzophenone.