I doubt that 30 year window. If if were nearly impossible to build a resistance, the bacterium producing the antibiotic would poison itself.

And I am not sure how you'd convince people in underdeveloped countries to use the new and expensive drug only.

The bacteria has had far more than 30 years to develop resistance, and it could do it in stages. We don't know how many other bacteria over the gigayears managed to poison itself.

True, but for one, the antibiotic will be produced and used in really large doses once it enters the market, meaning there are trillions of bacterial cells encountering it, which raises the likelihood of development of resistance. And second, horizontal gene transfer between bacteria means different strains of bacteria can cooperate in defeating it (https://en.wikipedia.org/wiki/Horizontal_gene_transfer)

I feel like reading about cryptanalaysis and entropy.

I'm not sure the resistance to other antibiotics would die out.

Resistances do die out; they typically aren't free to acquire. However, I suppose it's likely that a mere 30 years isn't enough to extinguish all traces, especially since it's impossible to really mandate a world-wide ban on certain antibiotics. So even if resistance is reduced, it's possible it'll to come back quickly.