ForestHillsFan, this is a service problem that varies branch to branch, and one machine being "24/7" doesn't guarantee the card reader, scale, or drop bin. I've used kiosks around the city for years. The best bet is to separate three needs: (1) buying/printing a label, (2) printing a prepaid return label or QR, and (3) dropping the item. If you need a stamped receipt, you want the kiosk or counter. If not, online label plus a staffed back counter until they close is usually faster.
To MurrayHill_M's point, buying a label online is often the only part that's reliable. For your location: the Grand Central retail floor is useful before 6; after 6, I'd try the Farley building. The main Farley lobby has kiosks and a self-service drop, but the bin near the 8th Ave side can fill up early evening when the Queens Midtown tunnel crowd hits. Don't rely on the last pickup time printed on the bin; if the bin is jammed, your package scans but tracking may not update until next day. I've learned to check the bin by opening the handle before paying for postage if possible.
I also wouldn't sleep on the smaller lobby at the Lenox Hill station on East 70th. It stays open later some nights, but the machine can be down too. Use the USPS locator and filter for self-service kiosk, then call before detouring, though that's a gamble. Last time the automated line said yes and the kiosk was off.
If this is for prepaid returns, the QR code return at a participating shipping store can be less stress. No receipt, but the drop scan happens in front of you. Not useful if you need proof of mailing. My practical order after work: buy label on phone, print at the office or library, then drop at a lobby with a staffed window still open. If I need a kiosk receipt, I go early enough the next morning and avoid the Saturday line entirely.