Right now there are still 29 million who are not covered and most of them probably never will be because it will be too expensive. And the current system doesn't cover everything for most people who are insured. Single-payer would cover everything from drug addiction to mental issues to cancer. How many families go bankrupt from having to deal with things like cancer?
And prescription drugs are too expensive, and currently Obama is trying to make that a reality all over the world, not just in the US, thanks to TPP and TTIP. Obama was also the one who banned Americans from buying cheaper meds from Canada to maintain Big Pharma's monopoly in the US.
Not sure I understand about going to a different healthcare provider. You can still go to a different hospital if one is too full.