Saying that a password manager is at risk to phishing is really saying that some users are at risk to phishing. It's not something that can be fixed. Users have to be vigilant.
If a password manager fails to fill in a form due to phishing, it won't prevent someone from opening the password manager and copying and pasting credentials after not knowing why the manager failed to do it on its own.
Anybody who tried to steal a password through phishing won't be targeting people who use password managers, but people in general. They will be relying on people reusing credentials across websites, which is something that won't happen to users of password managers.
It's not surprising that it's a low priority. It would be a lot harder to get people to install a rogue Gmail app than to simply try 12345 and seeing if it works.