What is needed to manage emails for groups? (replace Google Workspaces)

Question

This is a very newbie question, but I hope it's still acceptable.

Problem

My group uses Google Workspaces (we're a small NGO), but now Google wants to charge for it and we'd be paying thousands of dollars every year.

From Google Workpaces, we only need need an email manager(? I'm not being able to find the correct terms). That is, we only use the emails from our organization's domain (www.example.com). And we need a way to have centralized control over them (so that the administrator can, for example, reset passwords, create listservs or create new @example.com emails on the fly that don't cost $8 per user, per month).

TL;DR the problem is: we have a domain for our website (www.example.com) and we want to be able to have centralized control over all @example.com emails, so that the administrator can easily reset passwords, create custom email lists and create new @example.com emails for free (or very cheaply)

Solutions?

From searching the web, this is what I thought we could do, but I'm not sure I'm heading towards the right direction:

1. Sign up for business email hosting with some cheap host.
2. Create free Google accounts and migrate the old user data (e.g. user@example.com) to this new account (e.g., user@gmail.com).
3. Forward emails to the new Gmail account (user@gmail.com) and configure it to send emails as custom domain (as if it was sent from user@example.com).

The problem with this solution is that, although we might be able to backup all emails received (and perhaps sent?) from the synced new free Google account (user@gmail.com), we still don't have a way to reset passwords if we need to or have more control over the account. Someone can run away with the account (not likely, but a risk we don't want to face).

Question: Is there a web service that can manage emails, so we can create multiple @example.com emails for essentially nothing and preserve centralized administrative control over all @example.com email accounts? What are the correct terms for these things?

Related: G-Suite-like but in another platform like Microsoft?