Are loading times using a USB hard drive slower?
The external drive boots just fine to the login screen in 1-2 minutes.
  But then it becomes completely unresponsive for about 10-20 minutes
  after that (I can move the mouse around, but I can’t click or type
  anything).


10-20 minutes seems a bit long, but not far fetched; remember, you are actually connecting via USB 2.0 to a spinning hard drive. The White MacBook in 2009 came with a USB 2.0 port; even if you plug in a USB 3.0 drive, it will "fall back" to USB 2.0 speeds.

Maximum theoretical data transfer speeds:


USB 2.0 = 480 Mbit/s (60 MB/s)
USB 3.0 = 5 Gbit/s (625 MB/s)
USB 3.1 = 10 Gbit/s (1.21 GB/s)
SATA III = Up to 6 Gbit/s (750 MB/s) 
SATA II  = Up to 3 Gbits/s (300 MB/s)


USB 2.0 is more than 10x slower than SATA III and 5x slower than SATA II (3Gbits/s or 300MB/s).  If you have lots of things being loaded, it's going to take some time.


  ...since the internal hard drive is broken (it no longer shows up during
  boot or in Disk Utility


Why not get a new hard drive?  You can use any SATA III drive as it's backward compatible to SATA II.  In fact personally, I am using a Samsung 850 EVO in a White 2007 that runs a dual boot of OS X Lion and FreeBSD.  Any SATA II or SATA III drive will work. 

This is super easy to change out and only requires removing three screws in the battery compartment.  Ifixit.com has an excellent step by step tutorial.