Learn Finnish: Jopa kirjailija on kyllästynyt ideointiin. - Even the writer is fed up with brainstorming.

The adjective kyllästynyt (fed up / bored) has a specific rection: it requires the illative case for the thing you are fed up with. While in English you are fed up with something, in Finnish you are metaphorically fed up into it.

Does on kyllästynyt function as a verb in the perfect tense or as an adjective here?

It functions as an adjective describing the writer's current state ("is fed up"). However, structurally, kyllästynyt is the past active participle (often called the NUT-participle) of the verb kyllästyä (to get bored / tire of).

How is the noun ideointi formed?

It is derived from the verb ideoida (to brainstorm / generate ideas). By taking the verb's stem and adding the suffix -nti, you create a noun that describes the action or process of doing that verb. Here, it is put into the illative case to become ideointiin.

How would this change if the writer was fed up with a verb, like the act of writing (kirjoittaa)?

Because kyllästynyt requires the illative, a verb would need to be in the 3rd infinitive illative form (the -maan/-mään form). The sentence would become Jopa kirjailija on kyllästynyt kirjoittamaan (Even the writer is fed up with writing).