When I was a kid doing electronics as a hobby, I tinkered with circuitry based on vacuum tubes. I constructed circuits by doing point-to-point wiring between tube sockets and terminal strips. I used a soldering iron that looked more appropriate for soldering copper pipes. The components were usually mounted in a metal chassis, though sometimes I mounted them on a scrap of wood (the literal breadboarding of circuitry).
A typical breadboarded circuit.
(Source: Wikipedia)
There was no formal design or analysis process going on here. I was either copying someone else's design or tinkering just to see what would happen.
When I was a much younger engineer working with transistors, I typically fabricated circuitry on perf board using a point-to-point wiring technique. That's the phenolic board (like the inexpensive PC board material) with a grid pattern of holes across the entire surface, usually on centers of 0.156" or 0.1". Some amount of design was involved here -- simple calculations to set up a bias points for transistors, voltage dividers as part of a power supply, and power dissipation calculations in a Zener diode-based voltage reference. There was no computer simulation of the circuits ahead of time. I built it, double checked the connections, powered it up, and poked around with a scope or my Simpson 260 Volt-Ohmmeter. If I didn't like the performance, I would heat up the soldering iron again and make the needed changes.
A typical perf board circuit.
(Source: Wikipedia)
Following this era of prototyping, I moved on to plugging ICs and passive components into a solderless breadboard or plug board.
A typical plug board circuit.
(Source: Wikipedia)
There was still no circuit simulation, but at least it was much faster to experiment with different op-amps or component values and see the effect. The soldering iron could remain cold. Of course, these boards are trouble at frequencies above audio. Parasitic inductance and capacitance become increasingly troublesome. Further, at any frequency, contact resistance would cause reliability and noise problems.
Now that my prototyping involves surface mount devices, I am far more inclined to conduct a circuit simulation first. This is actually pretty easy now, since many IC vendors have versions of a Spice simulation tool available for free on their sites.
In the future, prototyping may get more difficult if vendors continue to shrink their package sizes. My guess is that reasonable package sizes will remain available indefinitely. I also expect simulation tools to improve. Expect them to be easier to set up to get the sort of results you need to see.
As more analog circuit functionality moves into integrated analog devices, circuit simulation will become all the more important. You will still need to prototype a design, but it may mean you have to spin a PC board to build your prototype, because plug board techniques will no longer be practical.
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