I've been following this thread as I own a 310A as well. I bought mine from my grandpa about 5 years ago, and he owned it for 15 before that so I know 20 years worth of history as well.
One thing we learned was to get rid of the single 12volt battery and put in two 6 volt batteries in series (6 volt, 3EH batteries). That made a huge improvement in cold cranking ability. The backhoe, when first purchased, had two 6 volt batteries, but they were shot. My grandpa replaced them with a single, large, 12 volt battery not wanting to mess with 6 volt batteries. The backhoe remained slow to crank and hard to start because of it. It had been overhauled when he bought it due to a cracked head, so knowing the engine was fresh, we just determined it was a hard starting engine and used ether. Flash forward 12 years, an old school mechanic friend of my grandpa's said to get rid of that 12 volt battery and put the two 6 volt batteries back in it. He did so and low and behold, the engine cranks faster. I've replaced them again last winter, but outside of some other self-induced issues I had going on, the backhoe starts quite well in all temperatures now.
Replace the coolant heater. Mine has an external coolant heater like yours does. Go to ebay.com, search external coolant heater, and buy one. They are cheap, $70 w/free shipping and when installed and working, do an excellent job in heating up an engine if everything else is correct in the system. I wouldn't waste too much time trying to fix your old block heater, mostly because your time has a value too. If you spend 5 hours on it and don't get anywhere, you wasted $100 minimum in labor and could have replaced it for less. In my case, I don't want to be wrenching on my hoe when I need it to run. YMMV.
Once you have a good set of batteries, a working block heater, and the main pump de-stroke solenoid worked out, you'll be very pleased with the starting ability of your 310A.
My procedure on cold days is to plug it in 2 hours before I plan on using it, or just plan on waiting 2 hours for the block heater to do it's job. Move throttle control lever to 1/2 throttle or full throttle (
@thepumpguysc just taught me this a month ago) and crank 'er up. I don't carry a can of ether anymore, it wasn't necessary. I feel dumb that it took me this long to learn this, but I caught on eventually.
Also,
@Delmer and
@GregsHD have been super helpful resources over the past few years for me in diagnosing and understanding my own issues with my backhoe. Thanks guys