Okay I have a Mazda 6...Mazda is something like 45% owned by Ford. It's an okay car, but definitely not my favorite (loved my Chevy Silverado 4x4, only issue was the paint job). Quality in American cars has improved drastically in the past couple of years. I believe Ford and Chevy have gotten some BETTER quality rankings than Toyota recently.
They (GM, Ford, etc) were heavily weighed down by legacy costs (pension, healthcare, etc). It wasn't really sustainable for line workers with a high school education to be making the same as a software engineer that spent $40k on an education. Even highly profitable companies such as IBM got out of the pension game years ago. I'm not saying that's right or preferable, it's just highly challenging to budget correctly for a pension plan given all the unknowns (market returns, increased life expectancy, etc). That meant they had to cut corners (quality) coupled with an outdated organizational structure lead to them digging their selves deeper and deeper into the hole until they went bankrupt. Ford might have avoided bankruptcy, but don't kid yourself...the union could have EASILY pushed them to it, but they saw what was happening to GM and realized they had to change as well.
That said you aren't going to have much luck buying a new car for $10k from Chevy or Ford. You're going to be better off going with a Kia or Hyundai or something like that if price is the most important factor (and if you want new).
Also most "American" cars have hardly any domestic parts in them. I think the percentage is roughly around 20/80, with 20% being American and 80% being foreign. About the only guarantee is it was assembled in US/Canada. That's better than nothing, but still many Hondas/Toyotas were assembled in the US as well. That's how they avoided massive tariffs on imports...