livewire78602,
ANIMATED SUSPENSION gave some good advice, though I would add that the reason you so easily get pinch harmonics with digital/solid state amps and pedals is precisely because those devices heavily compress the signal. Or, more precisely still, because those devices limit the signal. Many, if not most, solid state and digital compressors these days also include a limiter, which functions the opposite of a compressor.
Compressors do not allow the signal to spike above a certain threshold. Limiters do not allow the signal to dip below a certain threshold. This means that weakly executed pinch harmonics will still ring out loudly. This is why such devices can become a crutch to playing, and tube amps have a reputation for requiring better playing technique in order to make them sound their best.
Line 6 amps and POD's have copious amounts of gain, compression, and limiting. This is why the effortless harmonics on my POD XTLive don't necessarily translate to my Mesa Triple Rectifier quite so easily. A real amp makes you work for it, but rewards you with much better tone that Line6 just hasn't been able to duplicate yet.
Simply running your tube amp's gain higher and higher won't make up for a lack of proper technique. In fact, I'd suggest you try the opposite strategy, since it worked for me years ago. Try switching to the clean channel and, with no effects at all, play your pinch harmonics. At first, it may be frustratingly difficult. After a while, though, your technique will improve dramatically, and not just with pinch harmonics. Your intonation, fretting, phrasing and vibrato will all jump in quality. In short, you'll be a much better player, no matter what amp you're playing through.
