Colic is a symptom, not a disease. It just means the baby is crying a lot and nobody knows why.
Food allergies are one potential cause of colic. So is an immature digestive system (in particular, it takes a little while for the gut to get colonized by the normal flora that aid in digestion). The colic might have resolved because the baby was allergic to the old formula, or because after a while new gut flora finally allowed them to digest things properly, or they grew and gas pains were no longer as big a deal (babies grow really fast!).
The baby was born early. The mom did some research and found a more easily digested formula specifically designed for babies with digestive issues.
(Colic...) just means the baby is crying a lot and nobody knows why.
I would add that it means "and we don't care and we feel like doing nothing for a crying baby is fine once we've labeled it." It's a practice I wish would go die in a fire somewhere because babies labeled as "colicky" are routinely allowed to wail for an hour or more while no one makes any effort whatsoever to comfort them.
I'm glad they were able to sort it out so quickly.
I don't think it's so much that doctors don't care as that most cases spontaneously resolve faster than any person can systematically figure them out.
There's a psychological benefit to the parents -- they're often pretty stressed out and feeling like they're the problem. Putting on a white coat and a serious face while diagnosing the problem as intrinsic to the infant relieves the parents of the burden of thinking they're bad at their new permanent job, and probably helps bolster their waning ability to handle trying to comfort a crying infant without snapping and harming the baby or themselves. I don't think the idea is to leave the baby to wail as much as it is to get the parents to stop taking the wailing personally.
Like most of the paternalistic little white lies doctors deploy, it's not usually the appropriate treatment for patients who know better. Sadly some doctors never learn to distinguish these little psychological tricks (which are an important part of the medical art) from real knowledge about human biology. If they frustrate rather than comfort you, and your doctor can't or won't be straight with you, it's time to find a new one.
Overall, I really like your comments here and generally agree with you about what is going on.
The problem is that it actively fosters a callous and ignorant attitude. It would be much better to tell parents "Sometimes we don't know and won't figure it out because babies are growing and developing so fast that it's a really fast moving target. So sometimes they outgrow the problem before we can confidently identify the cause of the crying."
Telling parents that crying is normal and should simply be accepted breaks an important feedback system. Crying is always an indicator of distress. Sometimes it's mild distress that is readily resolved with a nap, but it is always an indicator of distress. No parent should ever be told otherwise while we try to find ways to help them calm down and so forth.
Parents I knew who accepted the colic explanation in infancy continued to be insensitive and ignorant parents at later stages of development as well. They didn't magically grow more sensitive, attentive to the needs of the child and proactive just because the child got older and learned to talk.
So I think the practice of telling parents it is colic and babies just cry sometimes is generally a bad education practice.
Food allergies are one potential cause of colic. So is an immature digestive system (in particular, it takes a little while for the gut to get colonized by the normal flora that aid in digestion). The colic might have resolved because the baby was allergic to the old formula, or because after a while new gut flora finally allowed them to digest things properly, or they grew and gas pains were no longer as big a deal (babies grow really fast!).