You asked, "Is anything changing in taxonomy to move towards monophyletic classifications?"
Taxonomists/systematists today (and really since Darwin) aim towards monophyletic taxa and there has been great progress. Check out the 'Tree of Life' webpages.
In many cases it is not known or agreed which taxa are most closely related- so instead of having a dichotomous (bi-furcating) tree you have comb like arrangements- this is especially true in the bacteria and many of the older groups where the divisions took place long ago but over a relatively short period of time.
You mention hair and pangolins and cetaceans. Pangolins and Cetaceans have all the characters of mammals- including hair. It might not be obvious to us, but they do at least have a small amount at some point in their life, or in the case of pangolins, most of the hair is modified into scale structures. Even if they did not have hair, they would still be mammals. In most cases of a lost character, the gene to make the character (e.g. hair) is still there, it is just not functioning the same as in related species. (A person born without a functioning brain is still a human even if he does not enjoy all the characteristics of being human.)
Birds are a monophyletic group- that means they come from a single species. If that 'first bird' species was a reptile (which all biologists now agree it was) then reptiles (without birds) are no longer a monophyletic group. It might be convenient for us to talk about reptiles (not including birds), but for biological purposes we are missing something predictive when we use non-monophyletic groups like reptiles.
Are we apes? It depends on how you define apes, but if you exclude humans from apes you do not have a monophyletic group--and you don't have a biologically meaningful group. We humans are animals, and we are apes. There is no problem biologically with saying that, though some people have a sociological problem with thinking it.
Giraffes know the importance of long necks. Giraffes are superior to all other animals in the sense that only they can easily get the tender leaves and only they can see the predators off in the distance. Obviously (if you are a giraffe), you are distinct, and can't be simply lumped with other ruminants. Humans apply a similar reasoning when they say they are not apes (or animals).
Progress in the understanding of relationships has been partially due to molecular data and computer programs that can help differentiate between apparent relationships due to convergence, and real relationships due to ancestry. Phylogenetic techniques also help discern similarities that are not taxonomically significant but may at first appear important- an example being '5 fingered' in mammals. If we look at the 'hands' of monkeys, humans, horses, cows, goats, and sheep, we might think that the 5 digits of the primates is something that makes them a monophyletic group. But this is the ancestral character- we need an outgroup to the mammals- and you will see that reptiles (meaning reptiles +birds!) have 5 digits. Thus 5 digits is not a character (by itself) that can make a group within mammals- but something derived from 5 digits- such as the cloven hoof of cows, pigs, deer & sheep can help in defining a monophyletic group.
I don't see controversy within the scientific community- there may be individuals that disagree but they are not looking at all the data. I do know some humans don't like the idea of being 'ape' or 'animal', but that has nothing to do with what we are. Biologically and chemically we are part of a group that includes other animals called apes and no other animals. We share a common ancestor with all other primates, with all other mammals, with all other animals.
Animals and fungi share a common ancestor, more recent than fungi and green plants. That makes us more closely related to fungi than we are to green plants. That does not make us a mushroom, but we are Opisthokonts!