How do we know loop currents exist?
Loop currents exist because we define them mathematically to exist.

There are certain things that we take as axiomatic in circuit analysis: That components (resistors, capacitors and inductors) are linear, that the total current at a circuit node must be zero, and that the voltage around a complete loop must add up to zero.

The fact that components are linear gives us the principle of superposition: The response of a component can be divided up into parts; these parts can analyzed independently and then combined at the end to give the overall response.

Loop analysis is just one way of dividing the current through a component into parts that are easier to analyze. The current in any particular loop can't in general be measured directly &mdash; it's just a mathematical abstraction. But if we use that abstraction in the right way, it tells us useful things about the overall circuit.