UPDATE to original issue: now it works. Short version: used Chrome on Windows instead of Mac browsers.
When I went to reproduce issue for tech support at my hosting provider, the Custom Header > Header Image > Select an image… feature worked after all.
The only difference was that I worked from a Windows browser (Win 7, Chrome Version 28.0.1500.95 m) whereas I tried earlier from Chrome, Firefox and Safari on two different Macs (MacOS X 10.8.4 “Mountain Lion”). I had also tried a Windows browser but now don’t remember which one and didn’t test as thoroughly on it anyway.
I will re-try on a Mac tonight. I don’t know whether to hope it will work or not…