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GH-36257: [CI][Dev][Archery] bot requires pygithub 1.59.0 or later (#36467)

### Rationale for this change

pygithub 1.59.0 or later doesn't accept empty string token.

### What changes are included in this PR?

Don't use an empty string for token.

Don't use deprecated `pygithub.Github(token)` API.

### Are these changes tested?

Yes.

### Are there any user-facing changes?

No.
* Closes: #36257

Authored-by: Sutou Kouhei <kou@clear-code.com>
Signed-off-by: Raúl Cumplido <raulcumplido@gmail.com>