I do personally regard national lab affiliation to be a meaningful quality signal -- at least strong enough that if I'm interested in the topic I'll read the actual paper, which isn't something I can say for a lot of stuff coming out of certain universities lately.

Another good reason to note the institution is that this is a demonstration that DOE Office of Science research funding is generating results.

To me the downside of marketing results aggressively to justify one’s funding is that the quality signal can become inversely correlated. Is this valuable research or is someone who knows to play the DOE game inflating a metric to secure their next round of funding and/or expand their turf? Does this negatively impact researchers who do not have access to a comparable marketing apparatus?

As far as "marketing," the labs are contractually required by the government to do this. Average 'impact rating' and so forth are part of the performance evaluations. As far as "playing the DOE game," there are a lot of voices in the critical path of getting significant funding from the Office of Science, many of them generally healthily skeptical. I'm not aware of very many charlatans achieving high-level management positions or controlling significant funding.

In the "Google researcher solves math problem" article, the researcher would have had to get permission to publish the work. If Google had wanted, they could have kept the result secret and used it internally.