With respect MMM, I don't think times have changed. Fast stream Civil Service was always ultra competitive. In my day there were 1000 applicants for 20 Foreign Office places. That said, in this era of job-insecurity I would not expect things to have got easier.
OP, when posting on MN I try very hard not to be negative. I wince when people run down a school, say, or University on the basis of a anecdote-based opinion or local prejudice, and wonder what people who have content DC in those schools/Universities think.
However.....sports science...think carefully.
We know a surprising number of DC who are going on to study sport science or plan to continue with high level sport at University, and have met a number of people working in different parts of the industry. Most appear to be making the right choices. Types include:
Bright DC who are good RG candidates for other subjects but who are also high level athletes (National Talent schemes if not National representation). Most seem to go to the US on sports scholarships, but some will read sports science at Birmingham, Loughborough, Bath or similar, as a way of keeping up their sport. One example is Paula Radcliffe, who apparently turned down a place at Cambridge to go to Loughborough. These are the sort of people who might get jobs in sports adminstration, and many employers will recognise the value of discipine and other attributes that flow from sports achievement (Civil Service, pilot, law conversion, whatever). Loughborough offers a good range of joint degrees (maths/sports science etc,) though with surprisingly high entry requirements.
DC who live for sport. I can think of one DC who had a ball at his minor public school. He almost certainly underperformed academically but played every sport going, and ended up as Head Boy. He is now reading sports science. I can see him either returning to his old school, or somewhere similar where his engagement and people skills will be welcomed, or perhaps the army. Think Prince Harry with better A levels and without Las Vegas.
DC who like sport and who have don't have a particuarly strong academic profile. There seem to be an awful lot of Universities offering sports science. Ones I am aware of include Roehampton, Hertfordshire, Bournmouth, Southampton Solent, and South Bank. Here I think we start getting into "media studies" territory. Will a degree be worth the three years of loans.
The problem here, and in general I would have thought that sports science is a better choice than media studies as it involves a degree of scientific rigour, is that there are lots of other routes into jobs in sports etc. So community football schemes are often run by people who have gone through the academies of the sponsoring Premiership Club. A swimmer who had had a role as coach adminstrator, may be in a better position to get the next job up than someone straight out of University. Ditto a personal trainer who has been working in a gym or leisure centre, or a peripatetic coach who have been working regularly with a private school. That said some employers (GLL, who run many of London's leisure centres is one) certainly used to say they would only recruit graduates. However these are tough and often poorly paid jobs. The hours are rough, you need to be able to manage staff and keep the public happy and the interface between commercial firms and the public sector bodies owning the facilities can be very political.
I would also add, and this is after speaking to a newly retired games teacher, is that sport is an interest you can grow out of as you get older. She said she had been very lucky that she had continued to enjjoy being out on cold wet pitches right up until her very active retirement. Most of her colleagues had had enough much earlier, but at a point when it was difficult to make a switch.
I hope this is not too negative. My gut feel, based on the number of DC we know studing the subject, is that supply of graduates will be higher than demand. Individual sports can be cliquey and tend to employ their own. Your DD should think quite hard about the sort of job she might want, in which sport, and who she will be competiting against for that job. She should consider carefully other advice about either gaining a specific skill, eg physio, which can be used either in sport of more universally, or taking a combined honours which keeps other doors open.