This doesn't matter at all, information asymmetry based extraction is still there.

The employer is buying labor in large quantities from many sellers, and because they don't appraise all units of labor individually (they can't), they are individually negotiating the least amount of money they can get away with. The fact that prices are concealed between these individual sellers (candidates/employers) is the only way this can work (information asymmetry).

When a buyer is shopping for a airplane ticket, they are shopping for the relatively same service from many providers, but they know these different prices and choose the cheapest (no information asymmetry). When, however, the airlines are using personalized pricing and the buyer doesn't know this (information asymmetry), the buyer is screwed over because they pay the maximum they would for the same flight, while the next guy is potentially getting a steep discount.

In both cases, if the information asymmetry is cured (as in sharing your salary or what money you paid for a ticket), the strategy no longer works because this changes the individual price points (people have huge loss aversion bias and they definitely react to the idea of being screwed over). The fact that they are buyers vs sellers in these transactions doesn't matter at all.

"Shopping for the best offer" depends on knowing what offers were available to you and the others to begin with.