There is a reason nobody designs PCBs that way any more (even if cost of labor was not considered). so, for the moment, i cannot think of any advantage of designing boards using that approach:
- slow (manual)
- no design rules check
- no vias (and no multilayers)
- no solder mask,
etc.

1. "Slow (manual)" - Well, speed does not seem a deterrent here. If the aim is to produce a replica of a vintage PCB, it may well take a long time to design the layout, what is more important is that the product is a replica. "Time" here is a one time investment.
2. "no design rules check" - The kind of PCB we are looking at here, i.e.. audio with signal frequencies not exceeding 20kHz, design rules checking is a luxury. Track densities is no where near that of a modern computer motherboard having track widths of 0.1 mm spaced 0.05 mm apart, so in present case, DRC can be done with the naked eyes! I have designed hundreds of (audio) PCB's manually and never faced any DRC problems.
3. "no vias (and no multilayers)" - A simple audio amp PCB does not require vias or multilayer boards. I agree it is much easier to design a multi layered PCB, but at the same time the fabrication of multi layered boards is more expensive and the end product is costlier.
4. " no solder mask" - Wrong. Solder masks can be placed on any type of boards, whether manually designed or computer designed.
In conclusion, I would say this is a special requirement and justifies special treatment, a little more time spent in designing a vintage PCB should not be an issue at all.

 

unfortunately you are part of a very small circle, looking at resurrecting relic that nobody else needs. and you do not need it either, you just like it for some esthetic (retro) look. you are choosing form over function.
you never faced DRC problems because you did not use DRC. after all audio designs are not exactly rocket science, in fact there is very little to worry about - we are talking about few kHz, not GHz, voltage and current are low etc. this is why every electronics tutorial and DIY beginner book starts with things like audio amp. and for same reason they do not include things like DSO, TDR, etc.

as suggested before, since time is not a restriction, nothing is in a way of doing PCB design the same way it was done in the past. optionally, one could develop own plugin that would transform geometric shapes into organic looking ones. see how plugins such as "rounded tracks", "teardrop pads" etc work. (example in post #15). one could create plugin that transforms any layer into artwork made of zones, and distribute points using spline. that would be neat programming challenge and i would not be surprised if there are more programmers who would take on this challenge than people that would use it.