Why can we not expand $(a+b)^n$ directly when $n$ is a fractional or negative index?

You can, in a way. The generalized binomial theorem affords a definition of $binom{n}{k}$, for $ninBbb C$ and integer $kge0$, such that$$(1+b/a)^n=sum_{kge0}binom{n}{k}(b/a)^k,$$or equivalently$$(a+b)^n=sum_kbinom{n}{k}a^{n-k}b^k,$$provided $|a|>|b|$. Note this modulus requirement prevents us exchanging $a,,b$ on the RHS, even though the LHS is symmetric. (Another issue with exchanging the exponents is that $binom{n}{k},,binom{n}{n-k}$ are in general no longer both defined, let alone equal, unless e make sure to write the definition of binomial coefficients in terms of Gamma functions rather than factorials & Pochhammer symbols.) Note also that our summing over all non-negative integers $k$ also holds when $n$ is a non-negative integer, because in that case any $k>n$ yields $binom{n}{k}=0$. This case also lets us drop the constraint $|a|>|b|$ altogether, so its presence when $n$ isn't a non-negative integer is very important.