I'LL ask the obvious, have you checked oil level? If it's low or dry, it's possible it's locked up. I haven't had any issues with mine so can't offer anything terribly insightful.
Something did just occur to me - the gear shift operates the clutch on the 4213 (same as the 3813) and there's a delay built into it, ie it delays engaging the clutch and that delay needs adjusting occasionally - it's supposed to be a 1 second delay, but mine had gone to 15 to 20 seconds, so when i'd shift gears it'd be that long before the tractor would move. It could easily be even more than that and possibly so far out of adjustment, it's infinity.
I'll look for my pix, if i can find them, but the adjustment is a 12mm nut - if you look under the right rear fender, over the right rear tire, at about the 1:00 O'clock position, you'll see a 12mm nut staring at you - it'll have a second 12mm nut under it. Loosen the locknut (the first one you see), and my memory is bad these days, but it should be tightening the 2nd 12mm nut shortens the wait time or delay time. If i can find my pix, i'll comeback and post them. I'd turn it a couple of turns and try starting the engine and shifting and wait a bit. What i said about the delay over time grows in length, i wasn't kidding. when i finally got off my *** and ordered a svc manual to see how to correct that delay, i felt i had time to read the ft page of the local newspaper before the tractor would move again. But it's entirely possible it's just delayed on engaging the clutch. It's not an intuitive feature on tractors that i know of.
the two 12mm nuts are ref #1 in the diagram below
and in case, the shifter adjustment pages. In the 2nd page, i've highlited the transmission filler plug with a red arrow - my tractor, after 20 years, the trans oil had turned to approx 60-70W gear oil, it was a synthetic oil and a golden honey color. The oil level should be right below that plug opening - i replaced mine with a true synthetic 40 wt oil, (i wasn't about to pay honda's $15 qt price for, iirc 7 qts. If you do need to add some oil, or choose to replace, be sure to use a true synthetic oil (not mobil one) - i say true synthetic as back around 2000, there was a scandal about mobile one and castrol's synthetic oils - it came out that mobile one had lobbied the petroleum industry association to have them change the industry definition of synthetic oil to let them offer their petroleum based oil (that somehow they had chemically altered the molecular structure) but it was still a petroleum based oil. I suspect or i'd be worried the seals in the trans would not be happy with petroleum based oil. fwiw
hope that helps