...would make the steem worth more and give everybody more value.
Theoretically, but that's not necessarily true. Burning doesn't create more demand and the supply is always increasing. Unless the burned STEEM is more than the amount of STEEM being created through inflation, then the overall supply will still rise. It all depends on whether or not demand can outpace the supply - and that demand depends on whether or not the Steem blockchain can grow its network for actual use, other than the STEEM token being bought and sold purely on speculation.
But even if a significant amount of STEEM is burned and it greatly outpaces supply, there are many other factors that go into crypto prices. Right now, it mostly just depends on how Steem/STEEM is viewed by the markets in general and how much new money is flowing into cryptocurrencies.
As far as advertising goes...
I think the best option overall would be companies buying stake and using it to upvote their own content as a form of advertising. There's no need for burning, there's no controversy about "having to see ads on my precious Steemit website," and the popularity/rewards of the advertiser would be open to community voting/consensus.
The second-best option would be to sell those highly-visible placements for blog posts on the various content pages. This would require an opt-in or opt-out for users if they don't want to see them, but it would also provide revenue via a marketplace for advertising, which could then be shared with the users of a given interface. (Each interface could have their own advertising/revenue model and split that revenue with those who use their interface.) This would provide an additional use case for the blockchain (if the auction could be implemented on it), it would create a more traditional marketplace for businesses and individuals (something that they could easily understand), and it would create a revenue and revenue-sharing model (which barely exists in the entire crypto space, especially at the potential scale).
Now that I've written that, I think the second option might be the better one. I guess it depends on what your goals are.
Yea, maybe, but burning would be completely neutral, and it would be a long term counter to the endless inflation. But let's see what people come up with.