Yes, that's so, at-willness of employment and unions are separate.
I nevertheless believe it comes from the same impulse, because, in general, the employer is in the stronger position than the employee. (Not always, of course, and it's certainly possible that some employers regret not having contracts with certain of their employees.)
Union rights and at-will employment are intimately connected.
What do people get fired for? It's rarely a true "performance" issue. Mostly, it's anything that the company judges to be against its interests. (In contemporary companies where managers can unilaterally fire people, it's anything that is judged to be against the manager's interest.) Anything that is perceived as disloyal, no matter how small, will usually lead to termination. That's how human societies tend to work, so it's not reasonable to expect that (absent legal incentives to operate differently) corporations would behave differently.
At-will employment, philosophically, trusts the company to pursue its own rational interests and gives it the right to terminate an employee if it believes retaining him to be outside of its interest. I don't fully agree with it, but that's what it is.
Collective bargaining is something that governments have decided that workers have the right to do, even though unionization is decidedly against management's interest.
The fundamental question here is: is it reasonable to expect companies to continue to employ people who, lawfully and ethically, act against the company's interest? There's no simple answer to that one. Greg Smith didn't do anything wrong, but I can't say Goldman would have been wrong to fire him (had he not resigned). On the other hand, the social benefit to unions (in theory, at least) and whistleblowing provides a good reason for the law to protect some categories of employees acting against employer interests.
Over time, the laissez-faire policy regarding employment relationships has been scaled back-- in the U.S., not enough; in many European countries, too much. It's now technically illegal to fire someone for attempting to organize a union, filing a discrimination or harassment claim, escalating a disagreement to HR, or even publicly disclosing one's salary (try that one in most white-collar contexts and see how long you last!) The problem with at-will employment is that companies have become very good at making illegal firings look "performance"-related.
I nevertheless believe it comes from the same impulse, because, in general, the employer is in the stronger position than the employee. (Not always, of course, and it's certainly possible that some employers regret not having contracts with certain of their employees.)