Let's consider teachers as an example. The top search result in Google tells me that 2% of the population of the US are teachers. With collective bargaining they can do work to rule, go on strike, and other job actions to pressure the government to improve working conditions. With voting they don't have a large enough voting block to push through candidates that will improve their working conditions. They would somehow need to convince over 48% of the population, assuming no gerrymandering, to also vote for the candidate that they want. That seems like a big thing to ask just to ensure an annual cost of living increase.
And given the limited number of candidates and the frequency with which our elections come down to a few percent, they certainly wouldn't have to sway 48% of the population to influence how elected representatives treat them.
You're not considering wedge issues? What if the candidate that would improve your working conditions also has a stance on an issue that you are strongly opposed to? For example, do you vote for the candidate who will give you a small raise but also disagrees with you on abortion? What if all of the candidates feel like siding with you would alienate more people than your voting bloc would bring?
> You're not considering wedge issues?
Maybe our systems of government shouldn't have this weakness. This is what I was talking about when I mentioned the "strong indictment".
Based on your last comment I think our disagreement is less about what makes a democracy and more about what constitutes collective bargaining. I don't see a difference between the examples you cited and a worker at a private company telling their neighbours to boycott the company until working conditions improve. Both of these are individual actions and unless they're organized at a larger scale the word "collective" doesn't really apply to either of them.
Examples of things other than direct voting include campaign contributions, talking to the media, going door to door. In the case of teachers, you could talk directly to parents during meetings.
There are all kinds of ways to influence elections that go well beyond simply casting your vote.
And yet, nobody considers that to be sufficient to deny workers their right to organize.
Board positions can be lifetime appointments.
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).
The reason it's strange is because if the government is a democracy (representative or direct), there should already be a mechanism for collective bargaining.
The fact that this is often insufficient strikes me as a pretty strong indictment of our systems of government.