Thanks for the info, @mmuller.
If hosting the meeting in your coordinator's absence is your primary need, you can use the Host Key method. It's pretty simple, but has a few requirements. This Zoom Support document has the details, including how to customize the key (Zoom creates a random key to start):
https://support.zoom.us/hc/en-us/articles/205172555-Using-your-host-key
This approach will only work if the meeting is set up with Waiting Room disabled, and Join before host enabled. With this combination of settings, anyone can enter the room (you can set how much time before the scheduled meeting the room can be entered (I recommend setting to 15 minutes if having conversations among early arrivers is a good thing, or 5 minutes if you need to restrict conversation.
At that point, you can join the meeting, and if the host has not yet started/joined the meeting, you have the option to "Claim Host" as described in the support document... basically clicking on Claim Host on the Participant panel and entering the Host Key. Voila - you're the host! If/when the coordinator joins, they have the option to reclaim Host, or remain as Co-Host.
If security of your meeting is an issue, either of you can enable the Waiting Room, and if there's someone that shouldn't be in the room yet, a Host can move anyone into the Waiting Room at any time (when it's enabled).
I do recommend changing the Host Key "regularly", like once a quarter if you two are the only ones with access to it, or once a month if you make the decision to share it with another (trusted) member of your organization.
That's the simplest method... but to your point about adding you as and admin: the non-profit admin can add your existing business account as an "external account" and assign you admin rights. This gives you additional capability to assist in managing the non-profit's account, but also has some additional "hoops" to jump through.
https://support.zoom.us/hc/en-us/articles/6583246047117-Managing-and-delegating-account-administration
(Ignore the reference to SDK accounts.)