I teach at a university and also dropped out due to homesickness on my first attempt.
Your daughter is absolutely doing the right thing. She needs to settle in and reminders of home make the homesickness pangs worse.
I promise you that I have almost every one of my first year students sobbing on me in a tutorial between weeks one and three. They arrive puffy eyed and their lower lip starts trembling and when I ask them to tell me what's upsetting them they always say the same, "I'm ok, I just really miss home/my mum/my dog" etc.
The ones that manage to distract themselves and keep busy and don't constantly keep going home or having visits/constant phone contact etc with parents do tend to settle much more quickly.
Some go home every single weekend, have parents visit mid week, or spend hours every night locked in their room facetiming home. They usually find returning after every visit traumatic and the start to every semester is miserable for them.
It's also these students that usually drop out.
I think it's why I dropped out, although the course and place of study wasn't right for me and I have no regrets. My parents panicked when I moved and didn't instantly love it and started to cry often and struggled to eat or sleep. There was no option of halls for me so I was living alone in a privately rented studio flat. My Mum decided she would stay with me to "settle me in" for a week. It made the moment of separation too drawn out. I think if she'd had left me the first night I'd have been forced to get on with things and developed coping skills more quickly. I decided to return home with her for the first weekend. Then my Dad drove me back on the Sunday night and stayed with me until the Tuesday. Then Mum came down on Thursday and we both went back for the weekend on Friday night and Dad drove me back on Sunday and the whole pattern repeated. It was a 6 hour drive. I was arriving home at 1am on a Friday as I didn't finish until 6.30pm. I was getting back to London at midnight on a Sunday. It was ridiculous, and largely instigated by my parents to "help me adjust". I don't blame them, they moved heaven and earth for me but it was unhelpful and they were babying me. Reading it back now, it was completely ridiculous. Very loving but kind of batshit and really motivated by the fact that they had empty nest syndrome and were sad I'd grown up and moved ao far away.
I always, always felt better on the Tuesday and Wednesday night when they were gone. I was forced to occupy myself. I could have a damn good cry and no one would know then I'd self soothe. I'd take myself off to a show or a museum. I'd organise my flat or do a pamper evening. I really missed home but when my parents weren't there I had to deal with it and find resilience.
Alot of homesickness is Separation Anxiety. The moment of separation is the most upsetting part and the more it happens the more difficult it is to cope with and the more the associated emotions and resulting behaviour becomes entrenched.
It's similar with little ones starting school or nursery. Often lots of tears, tantrums, distress at the gates or on the way to school but if you were a fly in the classroom just 20 minutes after drop off you'd find them happily playing with their peers having totally forgotten all about you. If you then went in to check that they were ok, the tears and clinging to Mum would probably start all over again.
I worked in a boarding school for a year too and again, the weekly boarders often had a harder time settling in than the international students who couldn't go home every weekend. Often a turning point for homesick boarders would be getting involved with weekend sports which would mean they couldn't go home. All of a sudden, we'd find they'd settled.
I know it's hard and completely understand you want to help her and also, you want to see her. But really at this point, you need to take her lead and trust that she knows what will work for her. I agree just a hug and lunch doesn't seem like too much or overbearing (nothing like my parents' overstepping!) but right now she feels it will make her worse. She can't bear the thought of separating from you again.
Like homesickness though, it will pass. At some point she will have settled enough that she will welcome and look forward to you visiting without any dread of the moment you have to say goodbye again.