feat: changed process.exits to throw new Error()s (#652)

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## PR Checklist

- [x] Addresses an existing open issue: fixes #538 
- [x] That issue was marked as [`status: accepting
prs`](https://github.com/JoshuaKGoldberg/template-typescript-node-package/issues?q=is%3Aopen+is%3Aissue+label%3A%22status%3A+accepting+prs%22)
- [x] Steps in
[CONTRIBUTING.md](https://github.com/JoshuaKGoldberg/template-typescript-node-package/blob/main/.github/CONTRIBUTING.md)
were taken

## Overview

Changed process.exit(1) to Throw new Error(...) in
src/shared/getOctokit.ts
Tried to do the same in src/shared/prompts.ts as suggested in issue
comments, but felt it was better with the original process.exit(1) code

Author: kristo-baricevic

src/shared/getOctokit.ts

@@ -25,8 +25,7 @@ export async function getOctokit(
 			console.error();
 			console.error(chalk.red((error as Error).message));
 			console.error();
-			// eslint-disable-next-line n/no-process-exit
-			process.exit(1);
+			throw new Error("GitHub authentication failed.");
 		}
 
 		const auth = (await $`gh auth token`).stdout.trim();