I am having to deal with this more and more these days. The solution is pretty simple, but it will take some time for you to see results.
The reality is that things are running and the business is functioning. Unless it looks like the business is going to stop functioning, you are not going to get any additional staff.
You have to build the business case for additional staff. Make a PowerPoint. I know it's lame, but it is all management understands.
Start the business case by first quantifying what is being done (basically the summary that you gave us). You detail how much time is being spent keeping things running. This is the Current State.
Then you have to quantify the additional projects in the pipeline. Just lay them out there. Do not judge them, just put it out there in black and white so that management understands exactly what is on your plate. This is the Future State.
THIS IS THE IMPORTANT PART -
You have to detail the risks of not hiring additional staff. Be honest, but not alarmist (unless you are 100% certain that what you are warning about will come true.) You have to lay it out to management that unless they do what you want (hire more staff), there will be consequences (the risks). It is then on them to decide whether or not they want to accept the potential consequences. These are the Risks.
There are two types of risks. There are risks to the future state, and there are risks to the current state. Depending on how dire you present the risks and the reality of them happening will shape the discussion.
The way they respond to an honest accounting of their risks will tell you all you need to know about your employers. If they take you seriously, they will either scale back the workload, or hire additional people. If they blow you off, you know nothing is going to change. It sucks, but at least you will have both eyes wide open.
You might not be the best person to present the business case. If not, partner with someone more senior who is in a position to have a frank discussion with senior management. Make sure that they are aligned with what you are trying to accomplish, otherwise they will undercut you.