There's no such thing as a remanufactured transmission, only a more thorough rebuild.
ALL of the hard parts get reused. A good, reputable transmission shop, they are going to use a master rebuild kit, in which case they replace every friction disc, friction plate, snap ring, gasket, and seal. No new bushings, no new bearings, no reshiming of the gearsets or differential. If a hard part is obviously damaged, they'll replace it.
A remanufacterer is going to do the same thing. The difference is that it's going to be done in a more controlled environment, with someone only rebuilding 62TEs and not whatever rolled into the shop. They'll spec and measure everything as they go, so the transmission will be reassembled to factory specifications instead of mechanic specifications.
I haven't worked on a 62TE, but I've rebuilt my 41TE and a Mazda transmission in the past. Mazda, it had dead friction disc's, no question. The three 41TEs I've had completely stripped apart, they all had hard part wear, and good friction discs. Two of them were rebuilt in the past though. The hard part that killed mine was the Overdrive Input Clutch Hub, and the Underdrive Input Clutch Hub was almost as bad. The 62TE shares the exact same part numbers for the input clutch hubs, I actually bought "62TE Input clutch" listed hubs for my 41TE.
Here's those 41te/62te hubs. Dead in my transmission 80k miles after a rebuild. Those teeth/splines are supposed to be smooth.
Here's a set out of the core that I rebuilt, which a shop had recently rebuilt, not much better.
The sad truth of transmission repair, is that statistically, one third of rebuilt transmissions in the USA fail before they leave the parking lot. It's cheaper to rebuild a bunch quickly, and eat the loss on units that fail, than to rebuild them thoroughly.
Make sure you do whatever you can to honor the warranty. You absolutely must replace the torque converter, and transmission cooler (which often means replacing the radiator). I'd go one step further, and use a torque wrench on the bellhousing bolts, then mark a line off the bolt head onto the engine block with a paint pen, that's top tier mechanic work and it'd impress me if 4 guys armed with beer and sandwiches in their garage were that thorough. Save the receipt for the transmission cooler, and radiator if it has a cooler in it. Clean out the cooler hoses thoroughly with compressed air I'd you reuse them, replace the hoses if you want to be safe.
The Torque Converter supplies the transmission cooler. The transmission cooler return is the supply for the lubrication circuit. If there is any trash in the torque converter or cooling system from when the old unit kicked the bucket, it will get pumped into the lubrication circuit of your new transmission, meaning your bearings and bushings get destroyed.