You assume that the entities doing physical shorting are the...

You assume that the entities doing physical shorting are the ones that are trying to do the takover (Goldman Sachs, City)

But normally, it's just another company doing the takeover (Company X)

Company X is usually acquiring shares in the company being shorted (Company S).

The lender of shares are usually big funds like Blackrock, Vanguard.

They don't have incentive to block low ball takeover of Company S by asking for shares back as they are also large shareholders in company X.
They are the ones that encouraged and orchestrated shorting through prime lenders (GS, City, etc)
If directors resist, they can put pressure by being large shareholders and trying to remove directors, force the matter to vote, etc.

Shorters usually cover just above the takeover bid offer by vacuuming up shares from holders. And a lot of holders are happy to sell as they think they are getting more than if the offer is accepted.
Shorts know in advance what the takeover bid offer will be, so they can work out strategy to make money.. Either on market or off market..

That's just 1 of the hostile takover strategies.

I don't think that's happening with GXY for the record.