worker: remove `api.BasePath` and replace with MakeHTTPErrorHandler() #17020

Job	Run time
⌨ Golang Lint	2m 56s
🛃 Unit tests	3m 41s
🗄 DB tests	56s
🐍 Lint python scripts	32s
🔍 Check for missing or unused runner repos	13s
🔍 Check for valid snapshot urls	18s
📦 Packit config lint	46s
🔍 Check source preparation	1m 6s
📦 RPMlint	1m 4s
🐚 Shellcheck	20s
🧹 cloud-cleaner-is-enabled	2s
Gitlab CI trigger helper	4s
🎀 kube-linter	14s
	12m 12s

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