I also have 2005-purchased LT1000 with the Briggs 31C707 engine. By early 2010s I also have issue of slow cranking, kept adjusting the valves, getting the battery checked or replaced, kept the battery on a charger.
Then in 2013, apparently while mowing the weeds in the pasture, I developed an oil leak somewhere and the oil leaked out, destroying the engine. I ended up buying a replacement 31P677 0912G5 engine online which was a couple more hp and also had a spin-on oil filter, and it was very easy to install. Since then, I no longer need to do the battery charger thing and have not needed to do a valve adjustment either. So basically the new engine took care of several issues.