I'm very curious why you don't feel any need to exterminate or modify predators, assuming it's likely to be something we can do in the future with some degree of caution and precision.

That sort of intervention is too far in the future for me to consider it worth thinking about. People of the future can take care of it then. That applies even if I'm one of those people of the far future (not that I expect to be). Future-me can deal with it, present-me doesn't care or need to care what future-me decides.

In contrast, smallpox, tuberculosis, cholera, and the like are worth exterminating now, because (a) unlike the beautiful big fierce animals, they're no loss in themselves, (b) it doesn't appear that their loss will disrupt any ecosystems we want to keep, and (c) we actually can do it here and now.