Studio engineers, do you check final mixes in lower quality headphones to ensure quality across price ranges? If not why?

Nope. The final is done on the best equipment available, and with bias minimized. Whatever cheap stuff the consumer has will come with its own bias that's totally unpredictable. Might be bass heavy, might be treble heavy, and I have no way of knowing. So, mix the recording to sound right on the most balanced equipment possible.