Why is SSH still asking for my passphrase on a private key provided by the server admin? (No passphrase set on private key)
Turns out I copied the data from the SSH key itself, but not the header/footer, and that was the entire issue. Note to self: make sure to copy everything from the start of the line BEGIN RSA PRIVATE KEY to the end of the line END RSA PRIVATE KEY including all dashes.

In other words, the private key is not...

4OoXTO0cqjtyd7rPWaKeKHgmF5Zu9h7AdZCIJuUTpbBhzER7nW1tMZVPAHMed3r
...
Z9BX7lae9FN/dv5TSAFdSG1Ow011ZvENw25l4mKMPREcHRvDMedeL6oQSBL2m9c


... but rather ...

-----BEGIN RSA PRIVATE KEY-----
4OoXTO0cqjtyd7rPWaKeKHgmF5Zu9h7AdZCIJuUTpbBhzER7nW1tMZVPAHMed3r
...
Z9BX7lae9FN/dv5TSAFdSG1Ow011ZvENw25l4mKMPREcHRvDMedeL6oQSBL2m9c
-----END RSA PRIVATE KEY-----