Hi Alan,
"Most 503 error links are from low quality directories, so I would disavow anyway. " Yes if they are low quality non-human edited then yes i'd disavow.
"We would disavow the majority of our links in one shot. Any risk of doing this?" If ranking is impacted by a toxic link profile then disavowing only 75% of them will not recover you 75%, probably nothing.
"Is there a reasonable chance that our ranking would improve significantly by disavowing these links? How long does it take Google to process the disavow? Is there a way of checking if Google has actually processed the disavow?" How long is a piece of string. The timeframe depends on how long it takes Google to crawl the toxic links.
Will this improve your rankings? I don't know is the simple answer. The best bet is to take the links on merit and disavow the ones you know are clearly toxic, manipulated etc. But soon as you mention improvement it makes me wonder if you have had a hit on organic traffic. If that is the case and it was around Sept onwards you may be looking at a broader E-A-T issue so disavowing would not resolve the bigger issue. That's pure guesswork but you get my point.
I don't know anyone who has any significant success with requesting links to be removed, other than sharks trying to charge to do so. You could argue that the 'good' sites will help, the poor sites ignore/charge, but it's a bit too much time and effort to use that signal in any way.
Mick