Please post the full, technical model number of your Craftsman. This number typically begins: 917. followed by 6 or 7 digits. This number is usually on the tractor body located under the tilting seat.
Knowing the full model number lets us view an online manual (if available for your model).
I will make an early presumption though.
If this tractor has a gear type transmission (manual gear selection) your drive belt may be stretched with age allowing an idler pulley to move too far when clutch is engaged, rubbing on something and making the noise you hear.

Or: It could have the same problem as the 3000 model i bought for $50. Failed transmission.
But, i was given an older Craftsman tractor with a locked up engine, yesterday, and today i checked it and--yep--it really locked up. So--guess where the good engine will be going---Yep--out of the red one, into the dark gray one. Parts is parts!

model #917.275810

If the tractor seems to be "pulling" OK, my original suggestion to suspect a worn,stretched belt can be disregarded.
After looking at the IPL for the transmission, I believe the grinding might be due to a worn/damaged needle bearing in the input shaft section of the transmission. The input shaft is what the transmission pulley attaches to.
The so-called "clutch" in this type of ground drive system is really nothing more than a movable idler pulley that either tenses the belt to feed power into the transmission, or loosens the belt to remove the power from feeding into the transmission.
But I don't suggest tearing into the transmission as your first step. Instead, check some other things first.
You are going to need good access under the tractor to shake, wiggle, and turn some components, so you need to remove the mower deck.
After the deck is set aside, you need to clean off the top areas of the transmission to remove any grass clippings and debris, so you might want to move the tractor to a spot away from your work area before blowing the tranny clean. Use a leaf blower or compressed air to remove the debris.
After cleaning, get under the tractor and take the ground drive belt off.
Grip the small idler pulleys (one pulley at a time) with your hand and try to wiggle the pulley as if you were trying to "break the pulley off its shaft". Do you feel any excessive looseness, or see a lot of "wobble" between the pulley and its shaft? If so, the pulley bearing/bushing is worn out. This could cause some grinding noise when belt is fully tensed.
Spin the pulley with a finger tip. Does it move freely? Grip the pulley with your hand and turn it back and forth on its shaft. Do you feel any rough "texture" sensation as it moves?
Do the same test on both small pulleys. If any/all the above conditions are noticed, replace the pulley.

Next, check the condition of the transmission input pulley and shaft.
Grip the edge of the transmission pulley with your hand and push up and down while watching what happens. Does the pulley tilt up on one side and down on the opposite side when you move your hand? If you see the opposite edges of the transmission pulley move up and down when your hand moves, the transmission pulley center hub is worn out, along with the input shaft splines too, most likely.
The next evaluation of the input pulley and shaft is to use your hand and push against one side of the pulley edge. In this, you push not up and down, but side to side, and front to back.
Watch the pulley while you push the pulley. Does it appear that the entire pulley and shaft move sideways when you are pushing back and forth. If so, the input shaft and the shaft needle bearings are worn out. Wear in these needle bearings and shaft can result in not just noise but also a rough vibration because the bevel gears at the bottom of the input shaft are operating out of proper alignment.

Here is a link that might be useful: 917.275810

Could be a bad idler pulley on the ground drive belt, being that it 'kind of goes away' when the clutch is depressed. You'll need to remove the belt from each pulley, spinning them and listening to each one for noise.