With the grain as in piece standing up on endgrain? With the grain as in sawmill-style where the log lies on the long grain?
Since I don't believe there's anything that will help the first be less than a stringy mess, and your mention of kerf-keepers suggests the latter, the answer is control. The mills use dogs and move the carriage rather than the log, which is what we should do in miniature. Such devices are made and commercially available at places like Woodcraft and Rockler. We can do the same with a "carriage" of plywood to hold a miter-groove referenced parallel. From there the plans get a bit more varied from wedges and screws to make our reference pass to moveable fences with clamps or dogs like a regular sawmill.
I own a scrub plane, so laying a branch in a set of v-blocks with a stop at one end (or your vise) allows a reference surface with minimum sweat if I didn't slab one with the chainsaw, which is easiest of all. Having this makes the round much easier to handle.
Second cut is the same as the sawmill, reference down, wedged for straight against a right-angled transport "L" and slab removed to gain the second face at 90 degrees to the first. After that, it's resaw work, but fewer teeth with bigger set is a good idea for any wet wood, and a periodic between-passes hand reverse of the blade through a WD40-soaked rag cleans up resin and acid sap that corrodes and ruins my edge.
Key for making circles is the same degree of control. Why a circle jig is the preferred method, as it keeps things at the proper position. If you're making rounds that you need to be perfect, the opposite face becomes important, so set it on your reference and stop-bore some reference depth holes on the DP to gain parallel.
