Nice. It does sound like a useful tool for sure.
If I was connecting to a business critical server, I'd want to do some analytics on the tool to make sure it isn't capturing any data or sending anything off to a 3rd party server. But that isn't hard to check, just time consuming.
My opinion, ideally you would have a rotating DBA schedule if you are needed 24/7. Probably cheaper to hire a second DBA to work the night shift than it is to pay you the overtime.
If they have specific production jobs that they need run after hours, it may not hurt to get them set up with permissions to run those jobs. Saves them the overhead and gives you a break from being on call 24/7. The blocking and low free space problems are a bit harder to troubleshoot automatically or let others handle, but I would hope that those problems are infrequent. especially low disk space.
My approach is to try to enable the users to work without needing to call in a DBA to do repeated tasks. My overtime bill should be as small as possible as DBA's are expensive enough on regular pay! Plus your sanity and peace of mind and sleep are far more important than work (my opinion). I would much rather have a nice supper with my family without being tied to my phone, or to enjoy some time on the patio without worrying that work will be calling and I will have to work after having a beer or 2.
My downtime is critical to my sanity and my health. The odd call for overtime is fine and expected and I get those. Heck, I have planned overtime one per month for roughly an hour to do some server maintenance. But unplanned overtime should be at a minimum, and I estimate that I am at maybe 4 hours unplanned overtime per year and 14 hours planned overtime per year. It used to be much much higher for both numbers, but it was starting to affect my sleep, my health, my friendships, and my relationship with my wife so I worked with the company to reduce the unplanned overtime. When I say it was much higher, it started at 9 hours per month of overtime when I first became a DBA here. I worked with my manager and was able to get that down to 4 hours per month which is the best we could do at the time without changing processes outside of our department. So we worked to changed those processes and now I am at 1 hour planned per month. I said 14 hours planned because sometimes that 1 hour runs into 2 depending on what is being done, but it is USUALLY 1 hour per month.
The older I get, the more I enjoy having time with my family and the less I want to do overtime for work. So if I can do somethign to reduce the overtime (such as building an application to allow unpriviledged users to run a specific SQL job), I will write that program and put the power into their hands instead of mine.