First, I'd make sure you are copying those full backups around to other places. If you lose one of them, you don't want to be going back a month or more. If you are going to let them keep you safe for 2 weeks, take extra precautions.

Can you use Service Broker? If so, forward notifications of issues through a queue to another server that can send you a note if you need to restart a backup when it fails. If you can let it go until the morning, don't worry so much about notifications.

I wouldn't use a central server to call all these jobs. If it has issues, and it will, you may end up missing notifications. The thing I've found when doing this is that at some point your central server will query server 1, then server 2, then fail. If you haven't accounted for that somehow, you might not realize that servers 3 and 4 were never checked.

The way I've done this is I have every server monitor itself. Make a small db (worst case, stick some tables in msdb) and query for backup status, jobs status each day. Store that information at each server. Make this a part of build status. I went so far as you also make a "report" table that merely stored the day's reports in a series of rows I could easily read with a query.

Then have a central server that has a list of all servers you need to monitor. Use that as a way to go get the report (or data) from each individual server. when you build your report, make sure you have a join with your central list and if there are missing servers, flag that.

I'd also make sure that I showed exceptions first, and then all the data. If you just show the data, it's easy to miss issues, especially over time and at scale.