Quick update - the device went back to Prof Tuners.
I emailed asking about "Linux support" and never got a reply. Firmware gleaned from a Russian forum and 3rd party forum isn't my idea of badged, marketed, advertised support. I persevered a bit more with different kernels, firmwares, etc, and then tried it on a Windows 7 desktop with their official drivers, recommended software, etc, and it displayed exactly the same behaviour. It would fail to tune to anything, and wouldn't pick any signals up off a known good source.
Emailed them asking to return it for a refund, I would pay return postage, etc, and by 2 weeks they never replied. Opened a case with Paypal, more weeks passed, and then Prof Tuners said via Paypal to return it within 6 working days... whilst I was on holiday over Christmas. Paypal kindly extended the returns period, I got it sent back, and then Prof Tuners seemed to refuse to believe it had been sent. I sent Paypal photos of the receipt, tracking number, and a screenshot showing it had been signed for and delivered to their address, and finally I got my money back from Paypal.
Clearly they make products that work, as evidenced in this forum. Maybe I got a broken one. Whatever happened, their customer service is appalling, and I won't hesitate in thoroughly recommending people to avoid them. Now that I have my money back, it's time to look at alternatives.
I really appreciate the time that people spent trying to help out! Many thanks for that, and apologies for not testing it on a verified (read: Windows) platform earlier.