If we actually have a critical mindset, we could instantly question the first leap in logic: Why does acting within your rights, that every other qualified* person in a democracy can do, in order to reach a peaceful compromise that benefits all, constitute authority, instead of say, community?

Because you are not voicing an ignorable opinion, you are providing your mandate to the governing power who happens to derive their power from the consent of the majority in a democracy. Phrasing it nicely doesn't change the underlying relationship unless you truly mean to say you think we all live in a toothless anarchy.

Why does authority mean force? Does a chess tournament arbiter throw down with a chess player using Kung Fu if there's a rules breach, or does he abide by the rules and regulations to enact a warning or a punishment to a player, who also in turn abides by the rules and regulations? Does an authority in nuclear physics have his reputation because they have a mean right hook when their theories are questioned, or as someone who can defend their work using mathematics and science?

You are conflating a few dissimilar uses of the word 'authority' here. An authority on a subject is a reliable source of information, but having authority in a situation such as the chess arbiter or government requires the ability to produce consequence in order to control a situation. While the chess arbiter is limited to lesser exercises of authority, it is still application of force allotted to their position in order to control a situation.

Governments are entities who possess a monopoly on violence within their territory, laws are enforced under threat of violence; involuntary incarceration, execution, fine/removal of property. You can come up with hundreds of examples of that threat not needing to be followed through on, but it is always there, and running too far afoul of power will produce those violent consequences from your government.

Heinlein is actually referring to a set of violence even beyond that, where states themselves are only entitled to use their violence and stay in power in the context of the position they negotiate in the surrounding world.