Systemic Error: Distributed Weight: Suppose you had a large loop, and decided to account for some of the lost length by measuring the height of the loop and adding that to L$.$ This would do a reasonable job reducing the error in L$$ for the fixed$-$length string, but there$$s still more issues for the stretchy string.$$

What would be the tension in the part of the string in the loop? $($Each string, not the total.$)$

Suppose we were to model the string as a spring $-$ i$.$e$.,$ the stretch is proportional to the tension. Why does your above conclusion indicate that, for a stretchy string, there is an error in L$$ from the loop that cannot be accounted for by just adding the appropriate extra string length?