You can add an electric starter to that engine. Main challenge will be finding a flywheel that will work. I expect the vertical shaft version (common on older Snapper riders) uses the same flywheel. That old Briggs is probably superior to a Predator.
That is the "classic" Briggs starter clutch. Here's a couple tips of how I work with them:
Briggs and Stratton sells a special tool to remove and install the clutch, since it doubles as the flywheel nut. However, here's what I do. First I dissassemble the clutch. Then I just take a socket extension and a hammer and place the big end of the extension on one of the parts that stick out around the edge of the clutch. Then I whack the extension with the hammer to loosen or tighten the nut. Kinda hard to describe. I reinstall the clutch casting to secure the flywheel before I assemble the clutch.
When reassembling the clutch, make sure you put plenty of oil on the skinny part at the front of the crankshaft or the clutch will squeal.
There should be six balls in the clutch. It's easier to put them back in if you lean the engine back.
After the clutch is removed I just whack the back of the flywheel with a hammer to remove. Works for me. Just don't whack the magnets.
By the way, why rebuild the engine? Does it burn oil?