Cool, It sound like you have fixed!
A binding pilot bearing can also spin the input shaft with the clutch pushed in. Sometimes the grease will get old and caked up in and cause the input shaft to still rotate with the engine when the clutch is released. A misaligned transmission (unlikely) could cause a bind between the pilot bearing and input shaft. Anything you can think of that would cause the input shaft to rotate with the clutch pushed in. There are only a few possibilities. Most of which have already been mentioned
I have had just about everything you can think of go wrong replacing clutches at work. Some of those phone guys would only get 2000 miles out of a clutch and back then almost every vehicle in the fleet was a manual trans. I would replace at least one clutch a week. Every once in awhile I would get a defective part or clutches that were boxed wrong. I learned to check everything first before that trans went back in.
First. Never ever let the parts store have your old core parts until you are done with the job and you no that everything is working perfect. If you give the old parts away and then you have a problem you have nothing to compare the new parts to.
Second. Make sure the linkage is good and has no play. Make sure the linkage parts are all correct. Where you get into problems is when you start building hot rods from twenty different year Ford's.
Third. You don't need to surface the flywheel every time you replace a clutch but its a good idea to do it. Its a judgement call and has a lot to do with who was driving the vehicle last. Any dark blue hot spots in the metal will grab the clutch disc differently than the surrounding metal. They can make the clutch grab real hard and jerky. Better to have them ground away. You should put a dial indicator on the flywheel surface and check the runout before you remove it. Have the flywheel machined, have that done first thing because it can take a couple days for a machine shop to do it. Make sure the crankshaft to flywheel mating surfaces are clean and then evenly torque that flywheel down using a torque wrench. Then check the runout again. While you have dial indicator on the flywheel you might want to check the crankshaft endplay
Third. I've seen people replace clutches but not the pilot bearings that was severely worn. I don't think they understand how important that little bushing is. Always, replace the clutch, the throwout bearing and pilot bearing together. Always make sure the new pilot bearing slides onto the input shaft and spins easily. If its snug your not going to be able to shift the trans. Its better to check it now than trying to put the trans in for an hour and then finding out the input shaft does not fit into the pilot bearing or the trans grinds with the clutch disengaged. I stick a little bit of disc brake caliper slide grease on the input shaft. Just a little , you don't want it getting slung all over your new clutch disc.
Fourth make sure the throwout bearing fits the front bearing retainer and slides easily. That there are no worn spots in the bearing retainer collar for the bearing to hang up on. We use to use white lithium grease where the throwout bearing slides on the collar but I think the synthetic caliper slide grease is better option. If its a flathead replace the return spring with a new one. In fact on any car replace the clutch return spring. they are always breaking.
Fifth. Stick the old clutch disc and the new disc on a flat surface next to each other and make sure they are the same height. They should be the same except for any clutch lining wear.
Sixth. Make sure the clutch disk fits on the transmission input shaft and slides real nice. Stick just a little of that caliper slide grease on the splines. Run the clutch disc back and fourth and remove any excess grease. You don't want the grease slinging off when its spinning.
Seventh. Make sure you have the clutch disc installed in the correct direction. On some clutch discs its impossible to install them backwards and on others it is. Use a clutch alignment tool to center the disc and then evenly tighten down the pressure plate bolts. Make sure you have the correct clutch pressure plate bolts. Make sure the alignment tool still slides in and out easily when you are done tightening the pressure plate. I use an old input shaft rather those wooden and plastic alignment tools.
Eighth. If you can, try to measure the pressure plate fingers to make sure they are even. On"normal" manual transmissions,not the flatheads I would adjust the linkage so the throwout bearing was tight up against the clutch fingers to hold it in place and centered. Otherwise they falloff or get misaligned when your trying to stuff the trans through them. The flathead throwout bearing design was brilliant. They should have kept it that way!
Ninth, If your having problems getting the trans to slip in stick it in gear and turn the tailshaft to line up the splines. At work when we had a stubborn trans, mostly on the F-600s trucks we would hook up a remote starter switch and spin the engine while somebody wiggled and pushed on the back of the trans. That usually worked pretty good.
Tenth. Stick the trans in high gear. Have somebody depress the clutch. We had a pedal jack to hold the pedal down so one person could to this. Spin the output shaft and see if it spins freely. That way you know the clutch is releasing before you put everything else together and then find out its not releasing.
Obviously you need to have the correct weight gear oil in your trans but the type of gear oil will not make any difference when trying to put the trans in gear when your stopped. That's because nothing in the transmission should be rotating.
As far as the clutch disc being to thick, it does happen. At work on a fleet vehicle that I had zero emotional attachment to. I would set the parking brake, put it in gear and then let the clutch out until it just started to rub wear away some of that fuzzy new lining. A couple of times it did actually work. I don't know what the flywheel and pressure plate looked like when I got done?