It may not be helpful to think of it as their brains turning off. Instead, perhaps their brains have never turned on!They may never have thought about what the operations there mean, and have been working on a stimulus-response basis where they see certain symbols and respond in certain ways. A combination of symbols that doesn't fit neatly into one of the categories they are familiar with will be responded to a fairly random way.
My instinct is that their expected response to an expression with a pronumeral in it is to give a new expression which still has a pronumeral in it, matching with their experience of expanding and/or simplifying. The answer being a plain number like 1 is not something they expect to happen and so it doesn't even occur to them as an option.
Also, their experience with pronumerals in the exponent is probably very limited, and I would hazard a guess that in all of those experiences, there was still a pronumeral in the exponent after simplification. So the pull to keeping the pronumeral in the exponent is probably strong.
They are probably not consciously seeing a pronumeral as a representation of a possible number. They see numbers and pronumerals as completely different entities with different rules of reckoning. All they have in their heads are rules for manipulating things. They don't realise that a good strategy to assess their understanding is to try it out with various numbers to see if it makes sense for each case, because they don't see it as a description of something that works for multiple different numbers -- it's just symbols.
To go further on this line, in one of your comments to another answer you say
... $p1^{p-1} = p^{p-1}$. It seems like there must be some reason that
nearly all my students think this same thing.
It is probably not true that the students think of these as the same thing at all. They probably don't even think of $3x^2$ and $3xx$, or $3 \times 1^2$ and $3$ as the same thing. All they see is manipulations that are or are not allowed and have expectations for what might or might not happen as a result.
In light of this discussion, you may need to really get them to talk about what they think expressions mean. Trying the expressions out with various different values of both x and p might help them to see that the expression is a description of how those numbers are combining to make an answer. I mention using different x's to highlight that 1 is a special case. Talking about how even if you don't know p, you can be sure the answer is 1 and how this would not be the case for most other x's would be helpful where the answer is expressable as a number, as opposed to not being able to do any manipulation.