"130 Desktops and max of 28 logical servers and you need 3 windows systems administrators!?"
Got to agree, I'm not even sure the guy is necessarily understaffed. When I worked IT support we managed a 6000 computer network across many different sites with 30 staff. Most recent places I've worked as a developer we had 2 IT folk for about 200 staff, 1 IT guy for about 150 staff (and client support on top). He's probably borderline where he is such that if a few major incidents occurred at once he'd really struggle, but that's something management have to accept. Depending on how important their IT infrastructure is to them then a day or so of network downtime whilst he catches up may be a non-issue to them.
It doesn't sound like they're understaffed on the developer side either frankly. It depends how big their applications are of course but 4 developers to support a 150 person what sounds like a manufacturing company is very much on the high side for that sort of industry. Maybe their development needs are exceptional, maybe not, but either way unless the management team has bought into the idea that their development needs are exceptional then they're not going to greenlight growing development any further. So as you say the developers either need to create a business case and come up with an explanation as to why the firms development needs are exceptional and what the benefits are, or they need to accept that they don't actually need more staff.
But this case sounds almost identical to a previous workplace of mine (where I was a developer) and having come from a larger IT support background I had a better understanding of IT support than my manager who was the IT support guy. The solution is to establish SLAs, get the board to decide on what response times they want and set up helpdesk ticketing software with those SLAs. If you continuously hit them then you were wrong about there being a problem, if you don't then one of two things can happen:
1) You'll get in trouble for not hitting SLAs
2) They'll recognise you need more help
If it's 2) then great, all is good. If it's 1) then this is for one of two reasons - either you're a slacker, and your bosses know it, or your bosses aren't reasonable enough to justify working for them anyway. Either way, if it's 1) whether you're lazy or not the fact is your bosses don't respect you and you're in a dead end job however you cut it. For my former boss it was very much the case that he resisted this idea precisely because he knew he was a lazy waste of space and he was only asking for more staff simply because he wanted to spend less time working and more time dicking around on the internet and whatever else.
So having SLAs set and a method of measuring them can be your enemy or friend in that they'll either make your case or get you in trouble, but if nothing else they'll let you know where you stand and make the decision for you as to whether you want to continue to work there.
It's only an anecdote, but in my experience "we need more staff" falls into three categories
1) You don't, you're just asking for your life to be made needlessly easier at the company's expense because you're lazy.
2) You do, and the company wont authorise it because they've no idea how to grow their business - get out now, you have no prospects there.
3) You work for a competent company where people are responsible and if you genuinely need more staff you don't even need to ask - you'll get authorisation for them as soon as you genuinely need them.
Either way the goal is to be the sort of person that fits into and gets a job at company 3).