feat: switch bin to Bingo's runTemplateCLI

[JoshuaKGoldberg]

PR Checklist

• Addresses an existing open issue: fixes 🚀 Feature: Switch to Bingo's new runTemplateCLI #1941
• That issue was marked as status: accepting prs
• Steps in CONTRIBUTING.md were taken

Overview

Corresponding PR to JoshuaKGoldberg/bingo#214.

🎁

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 95.39%. Comparing base (8f5b479) to head (3bfd41f).
Report is 4 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #1943      +/-   ##
==========================================
- Coverage   95.40%   95.39%   -0.01%     
==========================================
  Files          86       86              
  Lines        3522     3521       -1     
  Branches      201      201              
==========================================
- Hits         3360     3359       -1     
  Misses        162      162              

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