Seeking Allah's blessing (تبرك) through the person and possessions of His prophets is proven.
From the Quran regarding the use of the shirt of Joseph and the Ark of the Covenant:
اذهبوا بقميصي هذا فألقوه على وجه أبي يأت بصيرا
Take this, my shirt, and cast it over the face of my father; he will become seeing.
— Quran 12:93
أن يأتيكم التابوت فيه سكينة من ربكم وبقية مما ترك آل موسى وآل هارون
The chest will come to you in which is assurance from your Lord and a remnant of what the family of Moses and the family of Aaron had left
— Quran 2:248
Similarly from ahadith, where the Sahaba took blessing from whatever belonged to or touched the Prophet Muhammad ﷺ:
وإذا توضأ النبي صلى الله عليه وسلم كادوا يقتتلون على وضوئه
Whenever the Prophet (ﷺ) performed ablution, his companions were nearly fighting for the remains of the water.
فوالله ما تنخم رسول الله صلى الله عليه وسلم نخامة إلا وقعت في كف رجل منهم، فدلك بها وجهه وجلده
By Allah the Messenger of Allah (ﷺ) would not spit except that it would fall in the hand of one of his companions who would rub it on their face and skin
هذه جبة رسول الله صلى الله عليه وسلم ... وكان النبي صلى الله عليه وسلم يلبسها، فنحن نغسلها للمرضى يستشفى بها
[Asma bint Abi Bakr said:] This is the cloak of the Messenger of Allah ﷺ ... the Prophet ﷺ used to wear it, and now we wash it for the sick and seek healing thereby.
رأيت رسول الله صلى الله عليه وسلم والحلاق يحلقه، وأطاف به أصحابه، فما يريدون أن تقع شعرة إلا في يد رجل
I saw the Messenger of Allah when the barber was cutting his hair, and his Companions were walking around him, not wanting any hair to fall except into a man's hand
يا رسول الله، أنك تأتيني فتصلي في بيتي، فأتخذه مصلى، قال: فقال له رسول الله صلى الله عليه وسلم: سأفعل إن شاء الله
O Allah's Messenger (ﷺ)! I wish that you could come and pray in my house so that I may take it as a praying place. The Prophet (ﷺ) said, "Allah willing, I will do that."
And there is no difference of opinion on this.
It is not shirk because it is understood that:
- the benefit is from Allah alone
- the object is only a means of His blessing
- the blessing is subject to Allah's will, if He withholds it then the object will avail nothing
- the act has been approved and ordained by Allah
All of these are contrary to the beliefs of the polytheists of Jahiliyyah. They invented claims about Allah from their own imagination and desire while Allah had given no authority to them, and they believed that the object itself had powers independent of Allah, and they believed that if one god was angry at them then his rival deities or the magical powers of objects would protect them.
The same applies to amulets. Amulets are shirk only when the above beliefs are held about them, or when it relies on supplications and devotion to false deities. On the other hand amulets which uses Quranic verses or Islamic duas are not shirk, and after that permission on their use has been narrated from a group of the Sahaba and the scholars.