Learn Croatian: Gledamo televiziju putem satelita. - We watch television via satellite.

It follows the preposition putem (via / through), which always requires the genitive case. Since satelit is a regular masculine noun, it takes the -a ending in the genitive singular.

Could I use the preposition kroz here since it also translates to 'through'?

No, kroz is used for physical movement through a 3D space, like walking through a door. When you mean 'through' in the sense of a medium, method, or channel (like a broadcast or the internet), you must use the secondary preposition putem.

Is the preposition putem related to the noun put (way or path)?

Yes, it is literally the same word! The word putem started as the instrumental case of put (meaning 'by way of'). Over time, it became a fixed secondary preposition of its own, which now requires the following noun to be in the genitive case.

Just as a quick refresher, why is it televiziju ending in -u?

The word televizija is the direct object of the verb gledamo (we watch), so it must be in the accusative case. For feminine nouns ending in -a, the accusative ending is always -u.