I've seen several version of the Burke Class and it is quite an old frigate dating from the late 2100's but alway with the saucer and engines underneath layout. The engines chnage though and by TOS a small under saucer hangar "shed" had been added.
The saucer is very similar in style to the Baton Rouge Class cruisers.
Additions to the frigate inventory during the "4 Years War", which is probally the war McCoy refers to in TOS, resulted in the construction of the Chadwick and Buford friigate classes, both built using the same hull design. The obvious limitation of such an old design for such a limited small frigate was recognised. They are severly undergunned and underpowered by the 2240's when faced with superiour Klingon frigates.
During the "4 Years War" the Saladin Class destroyer USS Nelson NCC 546 and the scout USS Hamilcar were badly damaged.
The oportunity was taken to use the Nelson as a testbed for a proposed frigate upgrade. However, for several reasons, the project was not a success and Star Fleet went with a totally new design of frigates which resulted in the Coventry, Quillen, Miranda and Surya Class frigates.
The USS Hamilcar became the test bed for a new conccept in Warp Propulsion.... high speed cruising by switching between pairs of engines.
From this initial test bed runnig 4 x PB 32 Warp Engines, the Catchon, Chetah, Constellation and the TNG Galaxy class evoloved. The Klingons also "borrowed" and used this technology on several starships including the L7.
Other testbed Saladins include the USS Pompey NCC 506, which was converted into a succession of twin engined destroyer test bed configurations.