How was Alexanders army able to deal with the Persian archers?

[seleucid empire]

Furthermore, the formation of small (city) states* is favoured by the Greek topography, and heavy infantry is a typical military element of city states.

*Polis doesn't necessarily indicate a city.

This leads me to my own question.

The Greeks and Romans both had citizen armies which meant both nations were militaristic at heart.

So how come the Greeks never invented heavy sword armed infantry like the Romans and instead created the more complex phalanx? They seems to have skipped an easier option. Was it something to do with the fact that they didn't come in contact with the scutum shield?

And would the result have been the same at Thermopylae if a roman legion was there instead of the Greek phalanx