Then if no one thinks that Jesus only reigns because satan is bound then everyone must agree that Jesus still reigns after satan is released.
No one is arguing that Christ quits reigning when Satan is released. Christ's reign is not the issue.
The question is whether the saints continue to reign with Christ in the same sense throughout satan's little season.
If Revelation 20:4-6 means that the saints reign with Christ for a thousand years, then once those thousand years are completed, that particular period of the saints reigning with Christ has necessarily come to an end. That doesn't mean Christ ceases to reign---it means that the thousand year reign described in Revelation 20 has reached its completion.
That's why I think the Amil position would actually need to address something more specific than--Jesus still reigns after satan is released. Of course He does. The question is---what happens to the saints who are alive at the end of the millennium when satan is released?
If we were to construct a scenario where people are being saved during the latter part of the millennium and some of those people are still physically alive when satan's little season begins, then there would have to be some kind of transition. They would no longer be living under the conditions of the thousand year reign. satan would be released, the nations would once again be deceived, and the final period of testing would begin.
In that hypothetical scenario, perhaps the Amil position would provide a better framework---but only if it can explain how those who are alive during the final days of the millennium are subsequently tested during satan's little season.
One possibility would be that Satan's little season corresponds in some way to the great apostasy/falling away of 2 Thessalonians 2:3. I'm not saying that's necessarily the Amil interpretation. I'm simply saying that, if Amillennialism is correct, I'd expect it to provide a coherent explanation for how these pieces fit together.
That's ultimately my concern with the typical presentation of Amillennialism. Saying, Christ continues to reign after satan is released doesn't really resolve the issue, because I never argued that Christ stops reigning.
The question is whether the Biblical distinction between the thousand year reign, the release of satan, and the final testing/deception of the nations can be accounted for coherently.
As a Premil, I think Revelation 20 naturally gives us a chronological sequence:
satan is bound.
The saints reign with Christ for a thousand years.
The thousand years are completed.
satan is released for a little season.
satan deceives the nations.
The final rebellion occurs.
satan is finally judged.
So the burden isn't to prove that Christ stops reigning after the thousand years. The burden is to explain what changes when the thousand years end and satan is released.