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While Professor X could, my understanding is he typically doesn’t pay attention to random people’s thoughts

I think and argument could be made that his powers would work the same way the "normal" senses would. You wouldn't necessarily wake up when there's normal stimuli, but if there was a loud noise or a bright flash, you might wake up. Similarly, a would-be-assassin might have thoughts which would be alarming to the Professor, even unconsciously, while sleeping.

The point is made at least in the "X-Men: Days of Future Past" movie, where the McAvoy portrayed young professor keeps suppressing his powers with Hank's drug in order to silence the voices in his head — and to be able to walk.

My point being that I think that much like with hearing, our brains learn to ignore stimuli, rather than not actually hear it. The brain filters. So his power is rather a completely new sense, instead of something he directs at people and then gets access to. It's sort of always there, like noise, you just have to focus on the right sound.