The main problem, I think, is that the engines would always have to be going at the same speed as the treadmill - not significantly faster or it will drive off the front, and not significantly slower or it will be flung backwards. People are good at
compensating for treadmills' increasing speed, but as you can see at the end of the video, at some point the man has to go slower than the treadmill and he's thrown off. With a plane, there's less of an issue that the plane can't physically go fast enough (because if the treadmill can do it, the plane almost certainly can) but the plane will be thrown off in the same way if for whatever reason it does go slower than the treadmill. So the plane has to match its speed increases to the treadmill exactly, or almost exactly. At takeoff speed (which seems to be between 150 and 225mph depending on type of plane, according to Google), delays become significant.
Or do they?
Human treadmills are about double a running stride, so let's make our plane runway double the length of a 747 plus a bit, or 150m. We'll assume we've placed the plane exactly in the middle of the treadmill in case it goes too fast or too slow - ie, there's half a plane length in front and behind, 37.5m each way. 747s take off at 180mph, according to one source; if the plane is going at 170mph when the treadmill's reached 180mph (which seems reasonable), it has about 8.4 seconds to get up to 180mph - at that point, it will be thrown backwards off the end of the treadmill. (This, of course, assumes that it actually stays at 170mph for all of the 8.4 seconds- if it can reach, say, 175mph, while it isn't fast enough, it will have longer). The internet thinks that the 747 takes 75 seconds to get to 90m/s, so its average acceleration is 1.2 metres per second, per second. At 170mph, it's going at 76 metres per second, and 180mph is 80 metres per second, so it will take approximately 3.3 seconds to reach 180mph, which is safe, and it should take off.
If, however, the plane is only going at 160mph when the treadmill's reached 180mph - and if things fail, the plane would probably go a lot slower than 160mph - it has 4.19 seconds before it's thrown backwards (same assumption again that it stays at 160mph). Unfortunately, it has an acceleration to do which is supposed to take 7 seconds. Plane flies backwards and hits whatever's behind the treadmill.
It does occur to me that while it's financially unviable now, if land prices continue to skyrocket - no pun intended - giant treadmills might be cheaper than two-mile runways. But, as shown by the (incredibly simplified) maths, you shouldn't all rush out and buy stocks in treadmill runways just yet.