Excited has many meanings in English, from happy to agitated, and including aroused. The is no single French word that covers all of them so you should pick one of the meanings that best match the idea depending on what context you would use that verb in English.
- J’ai hâte de le voir would indeed translate “I’m excited to see him.”
Other translations might be
- Je suis content/impatient/enchanté/ravi de le voir
Je kiffe [de] le voir (slang)
Je (ne) tiens plus de le voir (colloquial)
Excité should not necessarily be ruled out, especially when “softened” by a preceeding tout :
- Je suis tout excité(e) à l’idée de le voir
While it is correct excité doesn’t properly translate excited when it is used to mean happy, enthusiastic about doing something, and in such case, people often make fun of TV interviews simultaneous (mis)translation, it is nevertheless exaggerated to reduce its French acception to sexual arousal.
“Excité” is quite common in French but limited to a real excitement. I would relate it to the idiom “je (ne) tiens pas/plus en place” meaning that you cannot stay still because you are too excited.
Je suis tellement excité(e) que je ne tiens plus en place !
Of course it can be interpreted in terms of being “sexually aroused” in some contexts but, at least in France, it is an advantage as we are really prone to make sexual jokes. And this not a troll, it truly is a way of life.