Battery cable, woven or solid

My 1954 L-21B has two (+/-) battery’s cables behind the passenger seat that connect to the battery box. Then two small woven cables connect to the battery. Lately I have been playing with the idea of having solid cables straight to the battery instead after having got problems with the connection and poor starting.

what is the advantage of having these small woven metal connections?

Electrically, it's okay to go straight to the battery. This is how Cessna did it. You'll need to remove the existing hardware from the battery box. Figure out how to install a rubber grommet at each cable location. If the box is tall enough, you can drill a hole and install the grommet. If it's not tall enough, you'll have to make a 'U' shaped hole and slide the grommet into the 'U'. Push the cable end through the grommet and connect to the battery terminal. When you install the box lid, make sure that nothing will touch the terminals in rough flight. Use phenolic, UHMW, or hard foam, as needed to keep the battery from moving.

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Boggert sells an adapter to do away with the braided cables. I'd ditch the G35 battery in favor of a Hawker SBS J16 with a .032"x 1/2" 4130 strap holding it to the side of the battery box.