For example an armed robber breaking into your house where you'd have enough time to broadcast that signed transaction (assuming you have wallet software handy and the tx was stored electronically!) but you would not have enough time to download Electrum, verify it, import your private key, fill in a receiving address that you have to recall from your head or notes, then then enter a password you made up to sign the transaction because you wouldn't put your cold storage at risk by not password-protecting a temporary wallet (right?) and finally broadcast the transaction before he comes to you and interrupts this demands at gunpoint that you send it all to him.
TBH, that's a pretty ridiculous scenario... I doubt the first thought in most peoples mind when an armed robber is attempting to break into their house is "I must broadcast that signed transaction I had laying around all these years and send those funds away".I would think it more likely that it would usually be "call the police"... or "get my gun/bat/knife/blunt object" etc.
Having unbroadcast, signed transactions should be a tool for "urgent" situations.