Is it a mistake (or trade-off) that size_t is unsigned?

Bjarne Stroustrup wrote in The C++ Programming Language:

The unsigned integer types are ideal for uses that treat storage as a
bit array. Using an unsigned instead of an int to gain one more bit to
represent positive integers is almost never a good idea. Attempts to
ensure that some values are positive by declaring variables unsigned
will typically be defeated by the implicit conversion rules.

size_t seems to be unsigned "to gain one more bit to represent positive integers". So was this a mistake (or trade-off), and if so, should we minimize use of it in our own code?

Another relevant article by Scott Meyers is here. To summarize, he recommends not using unsigned in interfaces, regardless of whether the value is always positive or not.

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