You could ask 50 SEOs this question and get 50 different answers - simply because Penguin case studies and definitive guides are pretty much non-existent.
Is it a cop-out to say: "You need to both?"
Part of me thinks: if you're under an algorithmic penalty - so long as the algorithm sees those bad links in your profile you will always have the algorithm's effect on the site. So by that logic, you'd want to remove all the bad links.
But then we need to remember - this is an algorithm. Has it seen some of the bad links but not classed them as "spammy/bad"? Those blog comments for instance, has the algorithm taken them into account? Or even registered them?
The point is that if you remove or disavow these spammy links, you could see your ranking decrease further, as they could be powering up your site still. With your traffic and keywords recovering somewhat, this might not be a thing you want to do.
If a gun was put to my head - I'd go on the notion that removing or disavowing all of the bad links would be the most definitive way of removing the algorithmic penalty completely from your site. However, all of those links would need to be replaced and I'm also living in a "perfect world" scenario. Many people are sceptical over whether a completely recovery is even possible and if using the disavow tool is pretty much like opening up to crimes before Google.
It's just so hard to judge. Looking at what's happening right now - I'd continue with the route of adding high quality and authoritative links to your site and monitor keyword and traffic progression. If it keeps recovering - great. If, eventually. the progression halts for a few months and won't improve no matter what you do - and you are not at a traffic or keyword level that you would be happy with - I'd then maybe look at the remaining bad links and say "you know what - you're holding us back now more than you (potentially) may be benefiting us. We're gonna get rid of you." Be completely thorough at this stage.
You'll probably see a rankings dip, but hopefully by resuming the healthy link building process you can reach new heights, which would hopefully vindicate the decision.
Hope this helps!