So you didn’t answer this question, but regardless …
Mail-in-a-Box is designed so that inexperienced people can host email and even a simple website all in one place. In your use case, you do not need all of that as you already have an established web site which may be a bit beyond “simple” as MiaB defines it. So you’ll want to keep hosting your web site where ever it is already being hosted, most likely.
There are two different ways to set up your DNS services. MiaB is engineered to handle DNS for you, so you could allow MiaB to handle DNS for your domain, OR you can continue to handle DNS at your registrar as you are currently doing. MiaB is very flexible in this regard. How you set your name servers is dependent upon this choice.
Now regarding your other questions:
Yes, absolutely as MiaB fully supports the major modern email standards of IMAP/SMTP/POP3. So you can literally use any mainstream email client.
Maintenance. You’ll need to routinely perform maintenance on your MiaB server. It is not much, mind you, but it is still another task that you should plan to spend a few minutes on every month.
Delivery reliability. Many VPS providers have low reputations for allowing spam from their network, so it may be a bit difficult finding the right one. Many providers also simply do not allow their servers to be used for email servers. Both of these issues can be solved with a SMTP relay though.
Cost. Hosting MiaB can cost from $4 to $10 a month.
I believe that it is 1GB ram and 1 core. The docs say (or used to say) less but Ubuntu, the required OS has become more bloated and will not install on a VPS with less than 1GB ram. I recommend at least 15gb storage though many use less. No less than 10gb though.
Now, back to the unanswered question:
If you are looking for privacy, MiaB is an excellent choice as generally only you would have access to your data. The downsides are cost which is minimal and the time and energy needed to maintain the installation, which again are minimal but are still a factor.
Let me know if you have any other questions.