It's very rare for care homes to give notice, it's hassle for them, it doesn't look good and it's something that the local authority/ CQC take an interest in when they inspect.
They are often very reluctant to take it as a course of action unless they feel they have no other choice.
Most of the time notice is given because the persons needs cannot safely be met and no action is being taken by the people who need to take the action (usually the NHS or the local authority but sometimes the family or the person themselves) so notice is given to force it to happen.
You shouldn't have been having to buy falls equipment, all of that is supplied by the NHS or the local authority even if you are self funding the care home placement.
It may well be that this care home isn't set up to cater for people with a regular, ongoing high falls risks.
Care homes can decide what level of need they are able to manage and it sounds like they have clearly communicated to you that they can't manage the risks your mum has now so she needs to move somewhere with a higher level of input.
Falls equipment does get moved (in the way) and tampered with (curious residents) and things like falls mattresses are also a trip hazard in their own right and should only be used after a proper OT assessment has been completed.
It may well be that the manager feels they have done everything they can to mitigate the risks and you have still complained.
or it may be they are crap and you need to report it.
either way it is very very common for people to have to move if their needs change and it sounds like that is the point you have reached.
It may be worth contacting the local authority for assessment for her next placement as a high falls risk may mean she is eligible for some NHS funding.