Breast milk nearly killed my little boy literally! They told me to keep persever that breast is best, even tho I knew it was making him ill.
Getting him admitted to hospital when he was 6 weeks old as he's bleeding from his anus was horrific, that was not to mention his breathing difficulties and skin sores...
His consultant was amazing and luckily my yellow book proved to my consultant my fears but they put me down as a stressed and anxious mother
soon we found out he had an allergy to casein.
Breastmilk stopped instantly and he was put on a extensively hydrolysed whey-based formula, but not until he had to have surgery on his bowel 
He was on that until he was about 4, until we could provide him with a balanced and nutritional diet (he has a limited diet due to other allergies) he's 7 now and a big strapping lad on the 99th centile 
He's actually considered obese which his consultant laughed at 
He's now the same height and shoe size of his 11 year old brother, however the 11 year old is dainty and the 7 year old is well built and in proportion, if he keeps growing the way he does his consultants think he will be 6ft 2-5 by the time he is a teenager, which is not surprising since our Dads where both over 6ft.
I honestly don't think it matters if you feed your child breast milk or formula, if they are going to be a sickly baby, they are and no amount of milk is going to change that no matter what form of milk it is.
My point being OP, they get all the illnesses when little, I think it's a mix of their vulnerable as babies and also having compromised immune systems, they won't forever be like this, they do outgrow it, even with ds2 having a compromised immune system, he is not getting ill all the time now, he's still picking up illness more that your average child but at least now it's lessening.