That's a good point. Gerry's aunts are definitely of the opinion that it's a shame and a pity when a young woman from a good family has to work support herself. On the other hand, the Trevennors, with 10 kids an a small income (five of them girls), aren't going to be able to support their daughters indefinitely (or at all after their death). But there's a big leap from a young woman going out to work as a governess or giving music lessons and going to university to train as a doctor. It's also odd that the offer is made for Jill (a girl), rather than the aunts offering to pay for the education of one of the older boys (who would need the profession to support a family).
I can see how the Trevennors might feel awkward at taking a relative (however distant) into their family and being paid to do so - the offer to pay for education is in part a face saving thing. Although I think the aunts would have had to pay for Gerry's school fees and later music education anyways, as the Trevennors were pretty stretched paying their own kids' school fees, and certainly couldn't afford Italian music training. So they'd be mostly be paying for Gerry's clothes and food and pocket money.
With the Chesters, all I can think is that Anne was in the depths of a depressive episode after Barbara's birth and all the other stuff, and not thinking rationally, and that Peter was going along to not upset her too much, figuring Beth's education was a reasonable sacrifice. After all, with looks like hers, she'd be likely to marry young to a wealthy man and not actually need a decent education.
Because it's a completely daft decision. They accept help for every single child except Beth, who has to go to a second rate school but is not allowed to socialize with her classmates. And Peter and Anne are totally oblivious to what it's doing to their daughter, and determined to nurture their pride at the expense of their daughter's emotional well being and future. Accept the offer for Beth's sake, and start preparing for a future where she'll need to get a job and support herself.