Would such Tibetan state be seen as "China" the way the Yuan Dynasty was? Or rather a Tibetan Empire in control of Western China, while an eastern Dynasty still survives? If contested you could have it so the Tibetan Dynasty then establishes itself in China proper, so even if the core of Tibet is pushed back the Tibetan's still hold power ala Yuan and more accurately the Qing and Manchu later on.
They'd need to take huge chunks of China, including more or less all of northern China. Otherwise they'd be more like the Liao or the Xi Xia, just ruling a part of China. I think they'd be more like a Tibetan Jin Dynasty than an early Qing, but I think they'd end up being seen as just as "Chinese" as the Jin were.
You do raise a good point that if the Tibetans are successful in China, they could reorient their empire away from India and Central Asia and toward China, abandoning territory to the Uyghurs, Caliphate, and other rivals and focus on ruling China. I don't see them conquering China, but they could end up the dominant state in Northern China while Southern China is ruled by a number of post-Tang states. They'd be pretty Sinified of course, but they'd still control the Tibetan Plateau.