I will attempt to address the following question only.
I'm asking whether the method of 'running' quantum algorithms on a 'quantum computer' 'simulated' on a classical computer would be able to outperform normal classical algorithms (preferably for problems that not obviously involve quantum simulation)
The closest thing to this that I am aware of are heuristic methods that employ natural computing, in particular the ones that take inspiration from quantum physics for the development of novel problem-solving techniques. These are known as quantum inspired algorithms. Please notice that: i) I do not claim that such methods could be rigorously shown to be superior to conventional algorithms, but it seems that they can be at least competitive; ii) the algorithms may or may not actually simulate a quantum computer, the faithfulness to the original source of inspiration varies.
I will briefly outline the framework of a particular type of a quantum-inspired evolutionary algorithm (QIEA). A more complete treatment may be found in chapter 24 of the book "Natural computing algorithms" by Anthony Brabazon et al [1]. Concrete examples can be found for example in arXiv.
The basic ingredients of a conventional evolutionary algorithm (EA) are a population of individuals $P(t)$, an update rule for the population, and a fitness function $f$. Here, each individual in $P(t)$ represents a possible solution to some problem, and $f$ quantifies how good the solution is. After initialization, for each step $t$ one evaluates $f$ on every individual in $P(t)$, records best ones and updates $P(t)$. This is iterated until a stopping criterion is reached, and the best found individual(s) are returned. In the simplest case, the update rule could be just random variation of individuals, but it can also be more complicated and engineered to introduce selection pressure towards better values of $f$.
In a QIEA, solutions are represented by bit strings of a fixed length, say, $m$. A quantum population $Q(t)$ is used, where each quantum individual consists of $m$ qubits. At each $t$, classical population $P(t)$ is determined from $Q(t)$ by "measuring" the qubits. $P(t)$ is ranked by $f$ and best results are recorded. $Q(t)$ is updated by acting on each qubit with a local gate, and iteration is continued. Often for $Q(0)$, all qubits are set to balanced superposition $(1/\sqrt{2},1/\sqrt{2})^T$, making each particular solution equally likely in the beginning.
As the quantum individuals remain essentially in a product state of $m$ qubits, there is no entanglement involved at any point, making QIEA not very quantum. On the other hand, we can effectively simulate the evolution of $Q(t)$ and make as many measurements as we want without needing extra qubits. The claimed advantage is over conventional EAs, based on supposedly needing fewer individuals or being better at maintaining diversity as the population evolves, as even a fixed $Q(t)$ can lead to many $P(t)$. All in all, QIEA by its design is meant to be run only as a simulation.
As a final remark, suppose that we wish to make QIEA more quantum without making it intractable. Can we? Perhaps. Consider the update rule of QIEA as a quantum circuit. It is rather boring, with a qubit register of size $m$ and a local gate acting once on each qubit. One could try to introduce some tractable quantumness to QIEA by taking the update circuit to be some Clifford quantum circuit, mentioned and outlined for example here and here. I do not know if this could offer any benefits at all, and as far as I know, this hasn't been tried.
[1] S. M. Anthony Brabazon, Michael O’Neill, Natural Computing Algorithms. Springer-Verlag Berlin
Heidelberg, 2015.
