Interesting question!
The reason for the restriction is simply to keep secp256k1_fe_set_int simple. Field elements are represented as 10 26-bit or 5 52-bit limbs internally, so restricting the function to only accept inputs that can be represented by a single limb means the function can just set all limbs to 0 and set the bottom one to the provided value.
While you're here, I'd like to ask a question about this format: On an initial inspection of the field_*_impl.h headers, I see that a single limb (depending on the field used), except for the biggest one, can represent up to 2^26 or 2^52 numbers respectively. @Coding Enthusiast actually mentioned that in the topic. There is also a magnitude for scaling the finite element up.
I'm certainly not asking for code to be modified for this, but I am just wondering - with these two facts, won't it be possible to set a larger range of numbers in only the bottom limb - while zeroing out the rest? Maybe it's because the representation is quite strange to me, and that I don't quite get how it represents 2^256 numbers with the limbs.