Much as I adore professional musicians they do have some very odd ideas about the best age to start various instruments, particularly as advice has changed even since I trained (not that long ago). They're often of the opinion you can 'never start too young' (wrong) or 'a teacher stifles creativity' (also wrong, jazz musicians are particularly bad for saying that for some reason).
They are right on some things - clarinet is a good precursor to sax - but very over-optimistic about starting at 7. Clarinet at 9 and sax at 11 I could buy into but both of them need your adult teeth to be firmly set. She could, as someone else said, start sax at 10 - my sister did as has had no problems that didn't exist before IYSWIM. You can get very good harnesses for sax that put the weight onto the shoulders and distribute it across the back. It's also worth noting the embouchure is not exactly the same and clarinet/sax players often have quite a 'tight' embouchure on sax and a looser one on clarinet. This could be a problem if she ever wanted to take up tenor/bass sax or Eb clarinet. With flute you don't get any embouchure 'confusion'.
Is she desperate to stay on a wind instrument? Can you bribe her with something like a guitar which you can't really do any harm fiddling around with? Lots of guitarists are self-taught.
A specialist teacher is an absolute must. In fact I would make it non-negotiable - she wants to learn sax or clarinet she goes to a teacher to do it properly. After a couple of years she should be sufficiently well established in her habits to start 'freestyling' but the foundation needs to be there. I play several instruments to above Grade 8 but I would only consider teaching those that I've actually been trained to do so. Instrumental teacher training has a lot about spotting and correcting bad habits that an excellent player won't even realise exist. Finally I don't think you can never start too late. I first picked up an oboe at 17 and did my Grade 8 3 years later with a good base from clarinet and flute and a healthy dose of motivation. I haven't played it since uni but that's not the point!
Digressing slightly and answering your original question:-
there are many makes of clarinet if you decide to go for one. It is better to get a good one that will take her to Grade 6 (Yamaha 3 series, Buffet B12 or B13) than a super cheap one that you will need to upgrade for her to take Grade 5. After Grade 6 you can take the decision to get a mid-range better model or a professional standard one. The brand you choose now in no way limits you later on. I had a Yamaha 3 series which took to G6. Very good intonation and Japanese manufacturing never let me down. They also get on well with poor(er) reeds, which is good when cost is a consideration and very forgiving for a beginner when you split a reed a week! At higher levels I've heard Yamahas compared to Hondas - always start in the morning but The One You've Always Searched For! I find with Buffet models the keys for the throat notes (G#/A) often stick and the quality seems to have gone down compared to the old Buffet B12s. I also recommend the Leblanc Vito 7212. Expensive and more of an intermediate instrument but sooooo easy to play is a Leblanc Bliss. I have an old series Leblanc Concerto (from before the factory burnt down and they did a total rebrand) and they're just divine to play. I've never had a go on one I didn't like. Personally I don't particularly like Selmer or Jupiter - Selmers are the exception the rule about moving on to a different brand when you upgrade (once a Selmer player, usually always a Selmer player)
I also wouldn't advise you to buy on ebay. Ideally you need someone to play it for you. This is where a teacher comes in handy as they can put you in contact with a higher-level student looking to upgrade or will know where to look. When buying check the pads, the movement of the keywork, whether there are any chips or cracks on the instrument itself and whether the keys are tarnished or flaking.
Finally it's worth investing in a new mouthpiece.