At the end of the day whitesands your daughter needs to come to terms with the situation, make her own decisions and take responsibility for her own application (and academic results), not lash out at you or resent the HofS . She would be very lucky if this is the only time she finds things not going her way, as I keep telling DD2 if she tries that on with real people in real life she isn't going to have many friends let alone a partner!!! It does sound as if her uni choice is determined by being at a RG in certain cities (let me guess big cities with good nightlife????) and if that is what she wants perhaps you should let her take her own risks, there are lots if things she can do to improve her application, reading around, getting involved in relevant extra curricular, talking to admissions tutors so she targets her applications /PS better (and maybe even makes an impression). Shouting at you isn't going to make any difference!!! Personally as soon as my DDs start that malarkey I walk away (doesn't stop DD2 trying though!!)
When you say that she was moaning at you that January was too late to put in her application, I would hope she knows that is rubbish, which suggests she possibly taking out her frustrations by winding you up. I do not believe any course at any uni runs out of the ability by by January, certainly at my uni it is part of the strategy for widening access, since late applications tend to come from students who don't have the advantage of being guided by good schools. Durham have had a policy of not making offers to anyone until Feb on some courses for some time for that reason, now obviously stretched to the Law course (glad to hear they now warn students upfront, has caused a lot of angst in the past)
On the other hand even if she makes one of her choices an insurance it is only 1 out of 5 options, it isn't going to weaken the applications to the other four and she still has four shots at what she wants. This year it is highly likely she will have better options at Plan B, HofS can only go on past experience. As creamteas says applying during a gap year has positive advantages.
Some unis, Bath, UCL certainly, have a sift based on points in which her GCSEs would score highly. Something to look into to assess the chances of success for her chosen courses.
Aiyah! not looking to DD2 doing this next year, as far as she is concerned it'll be 5 applications to Leeds, insurance will be studying the sociology of film studies ( not that I don't think Film Studies can be a rigourous and valid academic subject done properly, waves to wonderful Film Studies PhD across corridor, but you know what I mean)