What you are doing is redefining the words "government" and "market" to mean whatever you want them to mean and then rejecting them. I could redefine "auto-mobile" to be a subset of "dental operations" and reject it on the exact same logic.
Why not use the ordinary meanings of words?
And those ordinary meanings would be what...
Here's my simple take on it:
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Government1. (Government, Politics & Diplomacy) the exercise of political authority over the actions, affairs, etc., of a political unit, people, etc., as well as the performance of certain functions for this unit or body; the action of governing; political rule and administration
2. (Government, Politics & Diplomacy) the system or form by which a community, etc., is ruled: tyrannical government.
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If you break down the meaning of the definition into its component parts, you'll catch some of the keywords used to describe it. You'll find the words "exercise", "authority", "action", "function", "rule", and "administration". These words are applied to persons (aka. body, people, and political unit).
To "exercise" one's "authority", also known as "rule or administer" over another person (political unit), via some "function" is to "act" in such a way as to exert a force (to change the physical condition of an object). The object being person or property.
If you exert this force without my express prior permission (consent of the governed), or you do not act in such a way that would be construed or interpreted as to be in defense of your property or person from invasion, then you will have committed an aggression against me. This aggression or violent act would be what I would call a crime.
Now that we got the obvious out of the way, we can certainly see how a person or group of persons can decide either jointly or individually how they wish to defend themselves. They could do this by negotiating and purchasing the material means by which to protect themselves. These things originate from a market. I can make a market in guns, bodyguards, ramparts, forts, roads, planes, courts (arbitration), etc. These things don't spring up from nowhere. People have to make them, therefore markets emerge to meet the needs of the people. So yes, there is in fact a market for government goods and services.
Are we much clearer now as to why I used the definition I did?