Will a Tivo find and record streaming content?

This is probably a stupid question but I was wondering if one could set up a season pass for internet streaming programs. I'm seriously considering cutting the cord. Thinking of getting a Roku. I may be thinking old-school, as streaming is instant, but as I understand it, content comes and goes and if Tivo could capture "season passes" of streaming content that would be the cat's meow.

I watch very little major network programing so cost of an over-the-air Tivo and it's subscription price probably doesn't make sense.