You have it backwards. ISPs were trying to extort Netflix. They wanted Netflix, and every other CONTENT provider to pay them, the SERVICE providers, more money to have a higher priority connection. This means that, to use your examples, if Netflix or YouTube decided they didn't want to pay the ISPs an extra fee to have a faster lane, then YOUR viewing speeds would be lowered, and you would have no control over it.
Not only that, but they also wanted to do this on the consumer end as well. So this isn't all just happening in the "backend" as you put it. ISPs would be able to charge you, the consumer, extra fees as well in order to get the same speeds you already have. We've seen things akin to this happening already, with many of the big mobile phone service providers being scrutinized for throttling data speeds once people reach a certain data limit.
All of that is an extremely simplified version of how it affects you as an individual. On a broad scale, the internet is so integrated into modern society, especially in cities, that not having reasonable access to it would result in a significant lowering of living standards for the population in general. This is a big reason behind treating the internet as a utility and not an entirely privatized service. Another big reason is that in markets with incredibly high start-up costs and relatively low marginal cost of production, natural monopolies are a HUGE problem (lookup what happened with railroads, phone lines, etc for examples of this). It is very hard for new competition to enter the market, so once a few companies become established, if left unregulated they can make whatever rules they want and hardly anyone can do a thing about it.
This is just one side of the argument, but I hope this helped clear up the issue a bit for you :) I encourage you to look more in-depth into this debate and indeed all public/government matters. Knowledge is power!