My own dramatization aside, I didn't assume anything

It was spelled out in the articles I linked and cited that the companies involved did not think they would be affected by this pause, and they were moving forward with their planned projects.

Corpus Christi Caller Times: Cheniere taking Biden administration's LNG export terminal 'pause' in stride

As for the supply, the moderate Democrat that Republicans view as the ideal Democrat, Joe Manchin, intimated concerns about the coming supply.

Manchin, who almost never sees eye-to-eye with the administration on energy, said Thursday a pause could be appropriate if “done in a reasonable way.”

“If they do something reasonable that says, ‘We’re pausing right now before we continue to admit others,’” he said. “Well, fine, let’s put a time period on a pause. Let’s find where we are — what our reserves are, our production capacity and if we’re overcharging the market or not.”

The only people mentioning that it will cause an issue appear to be people who profit politically (politicians) or whose job it is to "protect" the O&G industry (trade groups). The Biden Administration was very clear that this policy could be reversed quickly if national security issues change. It's a permitting process that's one step in a very complex equation, and which occurs after other government approvals have already been received. As I cited in my post, two projects had permits expire because they hadn't even started construction. Expiring permits tells me the supply situation is not so dire that taking six to twelve months to update the data used in the approval process makes sense.

EDIT TO ADD: I also found it interesting that the pause announcement was made after the Brownsville, TX permits were approved. This article confirms there are four projects at the DOE approval stage which are affected (two of which were previously approved but the permits expired). These projects were listed under the Reuters article in my original post. Basically, any project that is ready to move forward was approved, further confirming that this pause will have negligible effect on export supply.

Another article laying out the facts:
Biden Administration Freezes LNG Export Approvals

As for the strawman, you might re-read my post. I didn't suggest those were arguments against. I suggested those were the first responses from the other side. My point was that when those responses seem to fit a little too cleanly then I'm naturally suspicious that they match the reality. As expected, in this situation, they don't fit the reality of the situation at all.