$2 post on the dealer button. Profitable to skip it, or worth it to play it?

I was playing 8-handed NL holdem yesterday, $3 SB, $5BB, and $2 posted on the button each round.

Twice over the 6 hour session, the dealer incorrectly moved the button past me and I skipped posting the $2 on the button (and got the CO position instead).

I was wondering, is this positive expectation play, to skip the button and the $2 post? Or is the button worth more than $2?

My take was it probably was not worth it to pay the $2 button...

I asked ChatGPT and this is what it said:

"In general, skipping the button and not posting the $2 can be advantageous in terms of profitability, but it depends on the specific dynamics of your game. Here are a few points to consider:

Position and Blind Structure: The button is a valuable position because it allows you to act last on each post-flop street, which gives you more information about your opponents' actions before you make your own decision. If you skip the button, you miss out on this positional advantage. The $2 posted on the button is usually a small price to pay for this strategic edge.

Cost-Benefit Analysis: The $2 posted on the button is relatively minor compared to the blinds, so in many cases, the cost might be justified by the positional advantage you gain. Skipping the button could mean you’re giving up the chance to be in a favorable position, which could potentially impact your profitability more than the $2 itself.

Game Dynamics: If the game has a tendency towards tight play or aggressive play, the value of the button could vary. In a tight game, positional advantage might be more significant, whereas in a loose game, the extra $2 might not make as big of a difference.

Short-Term vs. Long-Term: In the short term, skipping the button might seem like an immediate cost-saving move, but in the long term, the strategic value of being in position often outweighs the small blind cost.

Overall, while saving $2 might seem appealing, the strategic advantage of playing from the button position usually outweighs this minor cost. It’s generally better to post the $2 and take advantage of the position."