How do people estimate node voltage and branch current by observing a circuit?
A lot of it is simply seeing/solving circuits and subcircuits often, and being able to pull out those structures and consider their behavior in isolation. Even complex circuits tend to often decompose to a number of simpler elements, allowing a lot of behavior to be qualitatively described quickly, saving mathematical analysis for unfamiliar or unusual sub-structures.
These understandings of common subcircuits, of course, come from mathematically solving those subcircuits for behavior, typical operating point, input/output impedances, etc.
As an example for this circuit, one path of analysis that comes to mind is to:

Assume all MOSFETs in saturation.
Notice that Mk (likely strongly inverted) is giving a constant bias voltage to M3/M4 gates thus making M3/M4 matched cascodes for M1/M2.
Attempt to pattern-match M1/M2 to a current mirror, noting that the gates don't connect to the drain so I might not be able to immediately make that simplification.
Hand-wavily consider M1/M3 as if it were a single transistor with very high output impedance because a cascode structure behaves that way under the assumed conditions. Same for M2/M4.
Now the two sides match a current mirror structure, with high output impedance thanks to the cascode.

At each step, I redraw the circuit (mentally or otherwise) and continue making simplifications/equivalences that seem to simplify it further.
Knowing the circuit's rough structure and behavior allows me to decide what stimuli make sense for it--it can now be simulated to supplement and verify the analysis. If I get unexpected results, I revise my analysis, simulate again, and so on until my understanding and expected/actual results converge.