Why do certain companies keep paid content "accessible" through third party sites or addons?
Most major news sites, as well as some other sites of reading content like Medium, have a paywall for certain articles, but those are easily defeated by people who bother to search the internet.
As I suspect said companies are aware of that, and they don't react to properly protect their paid stuff, what do they expect to gain?
If you use a middleman such as archive.is it's not their bandwidth anymore but the middleman's. In most cases these services don't act as a proxy but store a backup of the article on their servers. Not sure if that's always the case though.
Those are typically explicitly allowed through for various reasons. They want people to pay, but they also don't want to stop Google/Bing and others from indexing it, and also archive sites. Which is why often people go through archive sites to bypass the paywalls, those can get a clean copy of the article and redistribute it.
It's not a big problem enough that they're probably deeming the loophole acceptable as most people still end up paying for it.
easily defeated by people who bother to search the internet
You overestimate the number of people who are savvy enough to bypass paywalls, or at least willing to put in the effort to find the content elsewhere . It's an insignificant number compared to the general population that it's probably more costly for them to do something about it rather than let it be.
Because fundamentally DRM doesn't work. It's effectively impossible to stop a determined attacker from gaining access to the information while also making it easy and convenient for the general public to access.
The point of pay walls is to be just annoying enough that 90% of the public go "screw it, have a few dollars", not to stop the 10% of people who were never going to pay you regardless.
Enough people want to pay for the convenience of getting all their news from one place so I guess they figure it doesn't really make that big of a difference