Can all "standard" properties of the tensor product be proven from the universal property? The tensor product is typically constructed in an existence proof by referring to a rather esoteric quotient space which "feels" hard to work with in general. The universal property of bilinear factorization feels much more natural and motivated - but is it sufficient to, for instance, prove that the tensor product is commutative or that a basis for $V\otimes W$ is $\{e_i\otimes f_i\}$ without referring to this construction?

Yes, of course. After all universal objects are unique up to canonical isomoprphism so you should be able to do all (or at least most) abstractly.
For example $V\otimes W\cong W\otimes V$ follows immediatly from $V\times W\cong W\times V$ via $(v,w)\mapsto(w,v)$, which respects bilinearity.
That the $v\otimes w$ span $V\otimes W$ follows from the fact that their span would also have the universal property, hence the inclusion of their span into $V\otimes W$ is the identiy.
And that any linear identity among $\{v_i\otimes w_j\}_{i,j}$ produces a linear identity among the $v_i$ or among the $w_j$ should also follow quite fast because then everything factors over the quotient
