I’ve never run a Grizzley lathe, but I’m unfortunately burdened at work with the Jet 1440 that’s a similar model.
The Jet has both a lead-screw and feed-screw disengagement lever. It doesn’t appear that model Grizzley does, but I would start with this lever:
Make sure it’s actually in mesh...
There’s a ton of slop in the gearbox levers on the Jet, I’ll assume the same for the Grizzley.
With the machine off and in a high gear, turn the chuck by hand and make sure all the levers on the gearbox are actually meshing. If the screw turns with the chuck, you’re done. Carry on with the break-in procedure and... “enjoy”?!?
Last possibility, other than Chinese “QC” would be the banjo:
If those two gears aren’t in mesh, you’ll never get anything transmitted to the feed/lead-screws.
I don’t wish to be too disheartening, but prepare yourself to be underwhelmed at best. My boss, and a friend, bought the Jet we use against mine and everyone else’s advice in 2017. To date:
The headstock seals are shot...
One of the Chinesium compound locking bolts literally vibrated all of its threads off...
I discovered there’s no drive-tang in the MT3 tailstock ram, so no tapping above 3/8”-16 in most materials.
That same tailstock ram appears to be made from the same material as the compound lock bolts... fortunately the drill chuck Jet supplies is even softer. Big-boy chucks and their arbors, not so much. Point being, you WILL eventually spin the MT3 shank in the ram and it WILL chew up the ram... QUICKLY!
The “lifetime”, sealed work light made it less than a year.
I gave up trying to level the machine, as the sheet metal stand deflects before you impart any force on the bed.
The 3 & 4-jaw chucks supplied are total garbage. The 3-jaw isn’t even heavy enough to make a good doorstop and I’ve found the 4-jaw has to be indicated EVERY TIME, but at least it actually holds material.
My boss liked that it was shiny and new... and it was for a month or so. Now it’s perpetually covered in oil, the paint is jumping off the bed, splash-guard and chip pan.
Best hope is that running this thing won’t put you off the craft completely and you’ll send it on on the way to a better machine soon enough.
Be safe and stay healthy
Jeremy