There is some truth (but not accuracy) to it, but not for this reason.
This article is about brainstem (aka hindbrain) vs the rest (midbrain + forebrain).
Evolutionally older animal lineages like fish, reptiles, birds all have midbrain + forebrain as well as the brainstem, so reptilian brain != brainstem.
Where there is some truth to our older "reptilian" brain is that the part of the forebrain called the pallium, that exists even in fish (& reptiles), has much expanded/evolved in mammals giving us our cortex and hippocampus, so you could kinda say that we have a reptilian brain + cortex. Note though that mammals didn't evolve from reptiles - they evolved in parallel from a common ancestor, so maybe better to say we have a fish brain + cortex (& hippocampus).
While the cortex is what gives mammals their intelligence, in birds the pallium evolved in a different way to give them theirs.
am no expert in brain development, still have read about evolution. I like aspects of the way you mentioned this, but seems that putting cortex separately from fish brain isn't quite right. As you mentioned, the cortex is simply a part of the fish brain that evolved much (much, much) larger than a fish's. Maybe:
Fish Brain = brain stem + small cortex (+ other parts of brain)
Human Brain = brain stem + gigantic cortex (+ other parts of brain)Sorry, know this is splitting hairs, but just thought it a useful distinction to make.