The reading passage argues that the Late Bronze Age Collapse can be explained by three compelling (competing) theories.

Compelling (interesting) in possible here, but since the theories can't all be true at once, I think you might have meant competing?

On the other hand, the lecturer opposes the ideas in the readingpassage by presenting several counterarguments.

First of all, the lecturer contends that a (climate change / a change in climate) is not the cause of the collapse.

climate change is a compound noun and also not countable. So you can say "climate change" in general, or "a change in climate" if you have one specific change in mind.

However, the climate at that time was relatively (more stable than before / stable compared to earlier times).

If the climate change is a factor, all civilizations should have been collapsed all together.

Secondly, the professor claims that pirates are not likely to be the cause of the phenomenaon.

Phenomena is plural, but we're only talking about one phenomenon here (civilization collapse), even if it's multiple occurrences of the phenomenon, they're all the same phenomenon.

Although it is true that the pirates attacked the region, they were not strong enough to wipe out the whole civilizations because they never worked together.

Removed the "the" as you're talking about non-specific pirates at the moment. If you wanted to talk about specific pirates instead, you would need to introduce them.

Finally, the speaker refutes the idea that unimbalance between the political and economic systems is not the reason for the collapse.

If he disputed that it is not the cause, that would mean he argued it was the cause.

The reason is that many other ancient civilizations experience thed instability too.

However, most of the civilizations with the unis kind of imbalance, survived for over a hundred years.

This goes against the writer’s view that the political and economic instability triggered the Late Bronze Age Collapse.

First of all, the lecturer contends that a climate change iwas not the cause of the collapse.

Since we are talking about long past events, it's best to use the past tense here.

He explains that it is true that a drought and a famine can end up withhave/cause devastating results.

However, the climate at that time was relatively more stable than it was before.

If the climate change is a factor, all civilizations should have been collapsed all together.

This is primarily because they never coordinated with each other.

Although it is true that the pirates attacked the region, they were not strong enough to wipe out the whole civilizations because they never worked together.

Finally, the speaker refutes the idea that unbalanceinstability between the political and economic systems is not the reason for the collapse.

The reason is that many other ancient civilizations experienced the instability too.

He mentions that instability of the political and economic systems of a society is a natural feature.

However, most of the civilizations with the unbalance,same volatility survived for over hundred years.