the youkai would survive due to the fact that people would fear them and their existence as they become visible to the world. You can say this is a cheap cope-out answer but I think the youkai would be annihalated by modern military capabilities. Even excluding the nuclear option what chance do most of these youkai have against the power of tanks, attack helicopters, tear gas and cluster bombs. Not to mention any illness that could fall upon them (of course this is assuming that youkai can be affected by human illnesses) due to the fact that the youkai have been isolated from the outside world for so long, new diseases could wipe them out easily. Now of course this only includes the low tier or mid tier youkai, F to B on tierlist terminology.
Some like Yukari would be much harder to simply take down with force. Some sort of a contract would have to be established with the world governments. Maybe this contract will be secret and this will be covered up as some massive government conspiracy where people are secretly fed to these creatures somewhere in some underground chamber. Possibly death row prisoners, people deemed persons of interest by the government. Although now this is starting to sound like the SCP foundation. Thats all the ideas ive got. plenty of people here will have actually good ideas about the subject matter im just some dumbass