Learn French: Fatigué, je veux juste cette friandise.

No. The phrase ne faire que (meaning "to do nothing but") must be followed by an infinitive verb, like je ne fais que manger (I do nothing but eat). Because cette friandise is a noun, you can't use ne faire que here. You would use juste or frame the verb with ne... que instead.

How would I use the restrictive ne... que in this sentence instead of juste?

You would place ne before the conjugated verb and que directly before the restricted noun: Je ne veux que cette friandise. This means exactly the same thing, but it sounds slightly more elegant or formal than using juste.

Why is fatigué placed at the very beginning of the sentence?

It is acting as an apposition, which works like a shortened clause meaning "Because I am tired...". Keep in mind that even though it sits at the start of the sentence, it still must agree with the subject je. If the speaker is female, it must be written as Fatiguée.

Why use the direct je veux here instead of the polite conditional je voudrais?

While je voudrais is standard for polite requests to other people (like ordering in a bakery), here the speaker is expressing an internal state of exhaustion and raw desire. The present tense je veux makes the statement feel more absolute, emphasizing their fatigue and singular focus on the treat.