First, there's really nothing small cost about Bannister's stopwatch:
https://www.alamy.com/stock-photo-the-omega-bannister-stopwa...
Sure, I can pick up the awesome Casio F-91W today for £10 and it would do much the same job. But for the modern day money equivalent of that gorgeous Omega piece I'm going to be able to afford a high-end GPS watch. There's nothing comparatively exponential about it.
Accuracy requirements will depend on training context - if you're running 60 second laps on a track and trying to shave 1 second of your mile PR like Bannister was, then you'll be wanting a certified track and decisecond accurate clock. If you're training for the marathon and running 5K repeats on the road, a few 10s of meters or seconds here or there doesn't really matter.
I can understand why some people don't want to run with a smartwatch, measure themselves, broadcast their progress to all on social media, judge themselves against others, and so on. (I also don't generally judge those who do, unless they're truly awful!)
I can also understand why some people would rather not run with any measure of time or distance at all. They feel it gets in the way, they'd rather just run free, they'd rather just run for fun, or they'd rather just race others for places in the spirit of pure competition.
I just can't get my head around the concept (and this isn't the first time I've heard it), that somehow old-school watches are acceptable but modern watches are bad. It smacks of neo-luddism (or sometimes hipsterism).