The on-chip memory will give about the same performance, with or without bursts. But I think that the SRIO core will slow down dramatically without bursts, as it will probably turn each read/write transaction into a SRIO packet. With bursts it would create less packets and I think that it would go faster.
The SGDMA is able to do larger transfers with bursts, up to 65536 bytes IIRC. But if you do a descriptor chain and regularly update it on the interrupt generated by the SGDMA, the software won't slow down the SGDMA controller, and it will run at almost 100% capacity even for big transfers.
The old DMA core seems very limited with bursts, it is a shame it isn't able to do a transfer bigger than the burst length...