I know Smith also wrote about moral instincts, but his point there was that those instincts are limited. We have more natural empathy for friends and family than for strangers.

And seizing on his invisible hand argument isn't necessarily the same as "turning Smith on his head." The problem is that the classic liberals wrote much less formally than the later, scientistic economists. The earlier ones were practically speaking off the cuff, so their philosophical remarks are subject to interpretation. Smith never developed his point about the invisible hand into a philosophical system or anything like that. His point was practically hyperbolic and rhetorical. Nevertheless, he said what he said, and his point was that even with our limited moral instincts and thus with our penchant for selfishness, capitalism could work, and it could work better than a command economy.