Linear circuits are mostly analyzed using the Laplace transform, i.e. in the s-domain, where the differential equations are abstracted away. In the time-domain, simulators are still used most often. But yes a really simple circuit like RC/RL is usually done on the back on the envelope, but then you're talking really simple.
The problem with the analytical approach to differential equations is that it doesn't scale well, and you don't know beforehand whether the approach will work, so you might as well use the numerical approach from the start.
Laplace transformations are differential equations so I fail to see your point. They're just in a different domain. However I do see your point with numerical methods since most complex problems are simulated anyways through simulating software. So in essence, the application becomes pointless because its at such a higher level of abstraction that you don't even have to think about it. You just punch in some numbers and hit analyze and the computer does it all for you.
> Laplace transformations are differential equations so I fail to see your point.
What I mean is that typically an electrical engineer will convert L and C elements to complex impedances (which depend on the frequency through s), and will then compute as though the elements are ordinary resistances. The expression "d/dt" isn't used in the entire analysis.
> the phasor transform thus allows the analysis (calculation) of the AC steady state of RLC circuits by solving simple algebraic equations (albeit with complex coefficients) in the phasor domain instead of solving differential equations (with real coefficients) in the time domain
This is like saying that if I convert mph to m/s then it's not a speed anymore. It's still a differential equation, just in a different domain because you can convert back from the s-domain into the time one.
The problem with the analytical approach to differential equations is that it doesn't scale well, and you don't know beforehand whether the approach will work, so you might as well use the numerical approach from the start.