Is fixing my iphone at a third party smart or will it hurt my device in any way

While I would only ever use Apple or an Authorized Service provider as sberman suggests, there are situations in which that might not be possible. If you have no other options, I can only recommend that you be very careful and be aware of the risks and downsides.

There are good third-party repair options, but you're on your own figuring out which ones those are. Even the good ones won't have access to OEM parts. It's also possible that if they work on your phone and you later try to have Apple fix it, Apple will refuse.

Start by contacting Apple Support. They can run a lot of diagnostics remotely, and I've never been charged for that. If you have an Apple Store nearby, make an appointment. The evaluation they do is free. Once they tell you what the problem is and what it will cost to fix it, you can make a much more informed choice.

If they are not an authorize service provider, it will render your phone untouchable by Apple.

Unauthorized service providers do not have legitimate access to genuine Apple parts or the tools required to properly install and register them.

TLDR: No. It is not smart.

If it's not an authorized service provider and If anything goes wrong Apple won't help you.

Also, unauthorized repairs can cause messages to pop up on the device and certain features be disabled when used or unverified parts are used.

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Only use Authorized Apple Service Centers if you don’t use Apple directly.

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"Is fixing my iphone at a third party smart or will it hurt my device in any way: Should I fix my iphone 15 pro max at a third party if I have no Applecare?"

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Getting This Serviced by a Third-Party:

That's YOUR call. But, you better read the Terms & Conditions this service provider hands you, before having them service it. If you do not agree to them, then I'd stay away. If nothing is offered to sign, then there may end up being no liabilities.

Which means they are authorized to provide out of warranty service and DO have access to genuine Apple parts, etc.

It's fine.