Yes making mass-emailing too easy is a footgun. It's incredibly easy to get blocked by common email providers entirely if you make a simple omission or mistake. Third party senders do a fair amount of work to ensure deliveability is maintained (because it's critical to their business). If you don't have a workflow to deal with spam reports and list management then you'll very likely be sending your emails into a void.

Spam has effectively ruined DIY mass-emailing, whether you like it or not. There are multiple other providers if you don't like mailgun.

There are a lot of ways for organizations and non-profits to save money, and I wouldn't recommend doing so with email if it's important to them.