Schools are in loco parentis. You are responsible for looking after the child while they are in your care. So yes, it is your responsibility to get a child to eat if they aren't eating well, because that's basic care. Primary school children are still very young, especially in KS1 and in early KS2.
Unless schools are going to let parents come into school to help their child at lunch, then yes staff should be doing that, it's all part of the job.
I went to a private school and our lunchtime was a set time when we all sat at long tables and ate at the same time. Once everyone was finished and the plates cleared, then we went out to play. Quick eaters learned to chat to others, slow ones learned to speed up a bit before the course was cleared. Each table was supervised by a teacher sitting and eating with us.
I think a similar system of a set eating time as a previous poster also suggested would help a lot. Fast eaters would know they had to sit and chat, and slow ones would know they had a set time in which to eat. Then everyone gets the same playtime.
My son was one of those who would not eat because he just wanted to go and play, ace I had to ask the school to monitor him a bit because he was starving and so angry by hometime as a result. Some days he was literally only eating a packet of crisps. If I only gave him crisps for lunch and no other food that would be seen as neglect, but people on this thread seen to think it's ok if it happens at school. It's baffling.
I despair of this attitude that children's basic human needs are inconveniencing teachers. Education is about much more than numeracy and literacy and if you don't know that then you shouldn't be teaching.