"because they are cutting costs by not hiring sufficient staff."

Or because the fire department has failed to implement something as basic as a call-out fee for frivolous calls. Of course a service is going to get over-subscribed and abused if the price for it is way too low. Both individuals and corporations will do that because it follows from the incentives provided to them. Lonely old people are going to sit on a public tram for 8 hours a day, taking up the space of legitimate travellers, because it's free to do so and the value they get from that is greater than $0. The same over-subscription is happening here with care homes.

If this is an ongoing problem and such a simple fix hasn't been implemented yet, the blame is solely with the incompetence of the fire authority for not having corrected the incentives with a call-out fee.