What are the compelling reasons for believing $L\neq P$?
Mulmuley's result (from Mulmuley's webpage without paywall) that, in the PRAM model without bit operations, "$\mathsf{P} \neq \mathsf{NC}$". (In the usual boolean model where $\mathsf{L}$ lives, $\mathsf{L} \subseteq \mathsf{NC}$.) This model is strong enough that the result implies any $\mathsf{L}$ algorithm for a $\mathsf{P}$-complete problem would have to look quite different from most known algorithms for $\mathsf{P}$-complete problems.

The PRAM model without bit operations is a nonuniform, algebraic model over $\mathbb{Z}$ (similar to algebraic computation trees or the Blum--Shub--Smale algebraic RAM model) in which the nonuniform program can depend not just on the number of integer inputs, but also on their total bitlength. In this way it's not a "purely" algebraic model, but lives somewhere between algebraic and boolean. This model includes poly-time algorithms for linear programming, maxflow, mincut, weighted spanning tree, shortest paths, and other combinatorial optimization problems, the logspace algorithm for tree isomorphism (see comments below), and algorithms for approximating the complex roots of polynomials, which is why I say any $\mathsf{L}$ algorithm for a $\mathsf{P}$-complete problem (which, as your question indicates you know, most people think does not exist) would have to look quite different from any of these.