meaning - Do "to infinitives" used with adjectives, such as "anxious", "unwilling", etc. indicate purpose?

"To infinitives" are often used as adjectives and adverbs to indicate purpose.

I have something to eat.

He came to play.

When we use "to infinitives" with adjectives like anxious and unwilling, are they said to indicate purpose, or do they indicate something else?

He was anxious to play.

He was unwilling to play.

This building is built to last.

Wikipedia's description says that "to infinitives" modify adjectives, but it doesn't say how it modifies the adjective. There seems to be a difference between a sentence like "he was unwilling to play" and "it is tall to reach fruit" in how the infinitives modifies, but if we say the infinitive in the first sentence does not indicate purpose, then I'm not sure what we would say it indicates. For example, when a "to infinitive" indicates purpose, it can be reworded to "in order to" or "for the purpose of" but I'm not sure how I would reword sentences like "he was anxious to play" or "he was unwilling to play."