As you said, it's more expensive. It's about 6 times the price for SRAM of the same capacity as DRAM. Of course, that only applies where SRAM can actually be manufactured in the same size. Die size for SRAM also grows quite large at higher capacities. It also consumes far more power.
If SRAM were used as the main memory of a PC, aside from the higher cost, the RAM that get's installed would be large enough that cases would need to be designed specifically to hold each SRAM Shelf that get's installed. Each SRAM shelf would need an effective cooling solution, equivalent to perhaps the air-channel-cowling used on graphics cards to keep the SRAM cool. It would also require a much more powerful power supply than is typically found in a PC.
it simply isn't worth the requirement to re-engineer a standard to accomodate RAM shelves rather than RAM sticks, and the added price of the power usage and cooling to use the higher speed SRAM as main memory.
Also much of the performance advantage from SRAM on the CPU die itself is a result of literally being physically close to the processor core. If the CPU had to reach across the bus for every memory access, it wouldn't matter how fast the memory is because the distance would impair performance below what the on-chip cache's currently have, and since a large majority of CPU memory access is able to access cached data acquired from main memory via the prefetcher, it would likely result in a net slowdown in performance.