@Beck: Some of the tape was new and some was used, but I was never able to detect a substantial difference between the new and used, in terms of dropouts. They pretty much all did it at some point during the first use. The only tape spool I bought that did not have a single dropout anywhere on it was actually a used partial roll. I think that one was Scotch, going by the brand on the reel, but that may not have even been the original reel. The new stuff was mostly Ampex 406, but there were a couple from a brand called AGFA. As for the age, I really don't know. I was under the impression that they don't make this stuff any more, so I just assumed most of the magnetic half inch tape in existence is probably pretty old. Any time I bought tape, I was kind of at the mercy of whatever I could find on ebay, and since the limited experiences I had with unused tape didn't indicate that it was any less likely to ruin the takes than a used one, I typically looked for whatever I could find that was used.
@lonewhitefly: I haven't used pure alcohol, but I have cleaned the head and tape path with regular diluted household alcohol several times, and it's typically only a couple uses before it starts acting up again. As for demagnetizing it, I got a new degaussing wand last year and used it once, but couldn't tell that it changed anything, really. I watched several tutorials on how to do it, and they all said the same thing so I assume I did it correctly.
@mdainsd: It wasn't the pinch roller. Looking at the front of the machine, it would be the very next roller to the right of the pinch roller. The first time I removed it to clean the melted rubber off of it a year ago, the threaded shaft on the inside of the cap (the part that screws the cap onto the roller) broke right off. Evidently, it was just plastic. I used alcohol to clean off the vast majority of the residue, and replaced it with a strip of adhesive foam rubber of the same width, like what you would find in the plumbing section at Home Depot. I had already checked out that Terry's Rubber Rollers page. That's where I learned how expensive the replacement would be. He has a picture of the kind I'm talking about on his page. It's priced at $85. $85 viewed as a normal cost for a random piece of equipment doesn't really seem like much if we're talking about an effects pedal, or a couple mic stands, or something that we know is going to do exactly what we want it to do before we buy it, but $85 for something that might not even fix the problem is a lot for someone on my income. Given that I've already tried all the other things suggested here, the next step would have to be taking it to St. Louis to get it properly serviced. That would be roughly $140 in gas, plus a minimum of $180 for the diagnostics which I was quoted over the phone, plus the $85 that I would have already paid for the roller that didn't eliminate the problem. That rounds out to about 60% of what I paid for the machine itself. You might call this a minor problem, but in terms of real dollars, the machine may as well be totaled. The majority of what I'm hearing from folks who are posting on this thread is that this type of thing is "routine" maintenance. If that's the case, it just doesn't seem to make any economic sense to me that I would have to routinely drop $400 on a machine that only cost $670.00. This machine has made better recordings than any other machine I ever used, but in the time I've had it, on unusable tape alone, I've already spent amounts equal to 30% of its value. Additionally, I had to pay $400 for a new mixer just to be able to record on it, since the inputs were all RCA size and the mixer I had before did not have individual direct outputs for each channel. That mixer stopped working about a year ago and the repair technician told me it wasn't fixable. I paid another $400 for a replacement mixer that only had half as many direct outputs as the first one, and now, I discover that I can't even get a single good take out of the machine because of this dropout thing. I have paid all that money and to this day, I've gotten about four usable sessions out of this machine. I'm sorry, but this is not an efficient use of money for a factory worker who has to support a family on only $40,000 a year.