I wonder if there's an inverse relationship between the level of pressure on scientists to publish, tweet, blog and demonstrate the social relevance of their work, and groundbreaking advances in fundamental physics.

I don't think so. The culture of publish or perish is certainly detestable and I don't think 160 characters are enough for a synopsis. But I don't think it negatively affects research. At least I hope so.

Fundamental research and humanities both have these funding issues. It may be worth a discussion if these fields could benefit from increased public funding, because some forms of research don't result in an immediate economic benefit.

I'd be really surprised if it didn't have at least some negative impact, not so much because of the amount of funding but because of what you have to do these days to get the funding. The more time and energy you force academics to spend on journal/conference submissions, grant proposals, impact assessments, and public outreach, the less they have to spend on research. I'd expect it to be especially true for more abstract subjects as they're harder to distill into a concise form or to describe the social impact of, and often require protracted periods of intense study/thought which admin responsibilities distract from.

Add to that the number of distractions that everyone is subjected to in the modern world, and I start to feel like it's no wonder we're still struggling with quantum interpretations or going beyond the standard model.

Now, I'm not saying that we should reduce the number of conferences or papers, or amount of outreach. Just the amount of time academics need to spend on them. Of course there will always be a few of the elite who will find time to sit and think for hours per day outside of these, nevertheless the harder you make it the more knowledge will suffer.

There was an opinion piece on PNAS a few years ago that argued just that, no hard evidence though:

https://www.pnas.org/content/113/34/9384

Thanks, that was a good read. Perelman is actually one of the examples I had at the back of my mind.

This is pretty much guaranteed to happen as explained by “the medium is the message”: https://en.m.wikipedia.org/wiki/The_medium_is_the_message