refactor(proxy): rely on auth header instead

[didinele]

Please describe the changes this PR makes and why it should be merged:
This more or less acts as a fix for auth: false being completely disregarded by the proxy if a client were to use it, which can run us into the nasty 401 bug.

🤖 Generated by Copilot at eb414a9

We are the proxy of doom, we forward the headers of fate
We don't need a token of lies, we break the chains of the gate
We handle the errors of wrath, we populate the response of hate
We fix the imports of death, we are the proxy of doom

Status and versioning classification:

• Code changes have been tested against the Discord API, or there are no code changes
• I know how to update typings and have done so, or typings don't need updating
• This PR changes the library's interface (methods or parameters added)

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2 Ignored Deployments

Name	Status	Preview	Comments	Updated (UTC)
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[codecov]

Codecov Report

Merging #9422 (cc939c3) into main (a2ab2b8) will decrease coverage by 0.02%.
The diff coverage is 64.10%.

@@            Coverage Diff             @@
##             main    #9422      +/-   ##
==========================================
- Coverage   59.31%   59.29%   -0.02%     
==========================================
  Files         222      222              
  Lines       14611    14624      +13     
  Branches     1254     1257       +3     
==========================================
+ Hits         8666     8671       +5     
- Misses       5905     5913       +8     
  Partials       40       40              

Flag	Coverage Δ
proxy	76.00% <64.10%> (-3.57%)	⬇️

Flags with carried forward coverage won't be shown. Click here to find out more.

Impacted Files	Coverage Δ
packages/proxy/src/util/responseHelpers.ts	81.70% <63.63%> (-8.46%)	⬇️
packages/proxy/src/handlers/proxyRequests.ts	77.04% <64.70%> (-1.22%)	⬇️

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[didinele]

sorry about that, sorta messed up a rebase.