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Hmmmm... Scratch head... Hmmmmm... Unfortunately, I'm not an expert on these tractors, I'm trying to figure out what might be happening based on my experience with my Cub Cadet 1811. I don't know the complete list of differences between the 1720 and the 1811 so some of my ideas may not apply to your tractor. Here's a couple more thoughts that I have had:
1. Follow the linkages from the Forward/Reverse handle back to the transmission. Somewhere in the linkage there will be a spring that isolates the transmission from the shift lever. The spring may be gone or the assembly might be worn out. Check out this website (follow the links for "more information" until you've read it all.):
Link 2. You stated that there is no transmission filter on your tractor. I don't know the Cub Cadet tractor line real well, but I've never seen a Cub Cadet hydro without a filter. Even if it doesn't have a filter, have you checked the fluid level and condition? It should be changed periodically because hydraulic fluid tends to absorb moisture and will eventually get enough crud built up to start causing problems. 3. You may want to check out the Cub Cadet forum on Yahoo groups. There are quite a few people there that know more than I do... Here's the link:
Link
It's free to join and you can learn a bunch! 4. Your brake problem may be independent of the forward/reverse problem. Do you have external or internal brakes? External brakes can be seen next to the rear tires. It's a setup that looks like disc brakes on a car. Internal brakes are inside the differential housing and can only be seen if you take the plate off of the differential. I would check the linkages from the brake pedal back to the brake assemblies to see if you can find a broken link or one that can be adjusted. Whew! I'm gettin' long winded here... Hope this helps!
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