"what about am/is/are? Is this for phonetic harmony? What about third person singular verbs?"
One kind of answer one could give to this is psycholinguistic: what use do people make of such aspects of language? The answer is: plenty, depending on the language; not so much for specifically am/is/are. During language processing, they are often used to inform sentence interpretation by telling you who does what to whom.
In the 70s and 80s, Bates & MacWhinney conducted extensive research on what kinds of cues speakers of various languages use to figure out what a sentence means. In English, word order is a highly reliable cue; what is directly in front of the verb tends to be the agent, what is directly after it, the patient (the situation is different in intransitive sentences, passive sentences ...).
- The cat eats the mouse.
- The mouse eats the cake.
The second sentence talks about a very dangerous mouse!
However, in languages such as Italian and German, phrases may shift around somewhat freely within a sentence. For example, it is possible to say both
- Der Junge isst den Kuchen.
- Den Kuchen isst der Junge.
Both of them mean that the cake (Kuchen, here in the accusative case) is eaten by the boy (Junge, in the nominative). Here it is case that tells us what is being eaten, and what is eating, not word order.
In other situations, case is a similarly unreliable cue in German. In these situations, agreement - the specific form of the verb, being dependent in form on its arguments - may be required to help you figure out who does what to whom. For example, English "is" tells you the subject is singular, so if you come across a plural noun, you know it cannot be the subject. For a German example:
- Ich weiß, dass die Frauen Peter gehasst hat.
- Ich weiß, dass die Frauen Peter gehasst haben.
Although verb order and overt case inflection are the same between those two, they mean different things; the first, that "women hate Peter", the second, that "Peter hates women". In the first one, the auxiliary "hat" is inflected to show that it agrees with a singular noun, Peter, so Peter must be the subject. In the second, "haben" must agree with a plural noun, Frauen.
Bates & MacWhinney demonstrated that native speakers of German rely mostly on case, speakers of English on position, speakers of Italian on agreement.
So in sum, in some languages, verb inflection can have very specific and important functions in sentence interpretation; in others, who often rely more on word order, not so much.
In the specific case of the English auxiliary suppletiva am/is/are, their status as morphologically marked seems to be somewhat of a historic artefact from a time when precursors of English relied more on inflection.
MacWhinney, B., Bates, E. A., & Kliegl, R. (1984). Cue validity and sentence interpretation in English, German, and Italian. Journal of Verbal Learning and Verbal Behavior, 23(2), 127–150. doi:10.1016/S0022-5371(84)90093-8