How to test for USB dropouts on STM32F?
Is there some method that could monitor a device for very long periods
  of time with millions of packets going by and find the source of the
  error?


Yes. The device is called "USB protocol analyzer". 

If you monitor only the host software side, the maximum you can see is that there was some "transaction error", and the port can or can't recover after dropping. USB protocol has hardware-assisted means to re-try failing transactions, and software doesn't have any visibility into "error count". So you need to identify the root cause of the error at physical level, on D+/D- wires. 

There are affordable USB analyzers, especially for USB 1.1 (FS 12 Mbps) rate. A good analyzer can be set for a sophisticated trigger while monitoring traffic in a long loop, or even recording the entire traffic up to capacity of your hard drive. I would recommend a small Teledyne/Lecroy model Mercury T2, but other guys like Ellisys and Totalphase Beagle are getting better and better.

However you need to be careful, since the analyzers are somewhat invasive, and their connectors/internals do have some effect on signal integrity. In case of flaky connection and rare error rate the analyzer can either improve the signal (and you might never see the problem) or can kill the link functionality (which will helpful to pinpoint the problem).

So in short you need to identify who is at fault when the device drop happens. It could be (a) device makes wrong responses to a valid USB protocol, (b) channel signal integrity problem, or (c) host hardware has a bug in handling some peculiarities of USB protocol.

I would start with (b) and check if all signals on the bus meet basic USB signal specifications: pattern frequency within 2000 ppm, jitter within the norm, signal edges are monotonic, and signal eye meet the diagram mask, all over your specific cables, devices, and hosts. There are standard procedures described in USB-IF website how to perform the electrical tests within USB compliance program.

If the signals meet basic FS signal specifications, the Protocol analyzer would be the next thing to deploy. If might be challenging to set up proper trigger and have correct interpretation of bus events leading to error. If you don't have experience with USB analyzers, you might need to take some training or get a consultant.