In a private company, there is also a mechanism for collective bargaining, without a union. The workers could just buy up all the outstanding stock, and replace the board.
And yet, nobody considers that to be sufficient to deny workers their right to organize.
That's generally only possible in a public company ("public" meaning "publicly traded"). Even then, I don't think the stockholders are under any obligation to sell to you, so you still may not be able to purchase a controlling interest in the company. So if the board holds a controlling interest, you're out of luck.
This feels like quibbling. In both cases, the mechanism proposed as an alternative to collective bargaining is wildly unrealistic.
With a corporation, you need to buy a controlling interest of shares. With the "voting as collective bargaining" approach for public sector employees, you have a chance once in a while (depends where you live, but 4-5 years in most places) to try and convince a plurality of the overall population that the concerns of your profession outweigh the many, many other considerations people have in choosing who they vote for.
Either way, both options are not realistically achievable by nearly any union (ironically, I could see police as one exception - "tough on crime" policies, which are usually favourable to police unions, tend to be popular among voters, at least up until June 2020).
With voting, you don't actually have to convince a plurality. You only have to convince a small percentage of the population (often <5%) to sway an election.
And yet, nobody considers that to be sufficient to deny workers their right to organize.