I know of nothing, including the power grid and Google's home page, which works 'seamlessly'. If you live in the Caribbean and there's a hurricane then it will take extra time to process your ILL request, just like your power or internet service may go down. But OCLC Resource Sharing is used in 49 countries, so while it's not perfect, it is solidly present even in 'third world countries.'
Yes, ILL works through public libraries and international public libraries. I volunteered with ILL in Santa Fe, NM and handled requests from Central America. I now live in a small city in Sweden and the closest library to me - 4 blocks away - handles ILL requests. Which I've used.
But my point is that ILL was put into place to keep people from having to fly "to some obscure state University" to read someone's "magnum opus". Adding the internet doesn't mean that older solutions to the problem no longer exist.