I am not forgetting anything.
You are assuming that link exchange means going to a site where people are requesting link partners. That is just a plain stupid method to find link partners. Only an idiot would go to a link exchange site to look for link partners. It will just be filled with idiot IM'ers like you find on forums.
I'm not assuming that is what it means, It's what I've seen like I posted before and yeah, I agree with you, that's a pretty dumb way to do it, but there are sites like that on the web and people doing it that way.
And no, getting a link from a domain that doesn't have backlinks is not going to automatically be a low-quality link. It might not be as strong or powerful of a link, but it is not low quality just because there are no links pointing at it. That is just total misinformation there.
True, it doesn't automatically turn it into a low-quality, but if you consider that a bad backlink or low-quality one is the one coming from an untrusted site it could be considered like that. However, at this stage I believe it's best considered as "neutral".
Checking stuff like DA is not a very good way to evaluate links either. Moz's metrics are terrible. I have thrown nothing but spammy porn and penis enlargement links at domains just to test it and pumped them up to DA's over 50 in every single case.
Unfortunately DA can be easily manipulated like you just mentioned and I wouldn't rely on it, but it's always nice to see more metrics (not very trustworthy but still, another metric), the one that is most trusted right now is trust flow wich is why I mentioned it.
Domain age is pretty much not a factor either. Wouldn't waste my time looking at that.
I agree that it is really not a factor, but domain age would be mainly to determine how long has that site been around and if it could be trusted. After all there would be an archive backing this up, and what I check is the archive. I always check to see if the site has been a PBN before, casino or porn site (I stay away from these because they tend to be spammy in some way).
There are many ways to approach this and many ways to do SEO so these kind of talks are always good to have