Public Sector Potential
I don't know what lots of money is to you or what the cost of living is in your area but generalists are very common in public sector. Specifically in the EDU space I consider the money pretty good.
That being stated "problem solver" is still just a toe in the door to carving out a technology set. For example I currently admin NetApp SAN, Cisco UCS, vSphere, Citrix, Active Directory...etc. I am not an expert by any stretch in any one of those.
I can also say that in the K12 space that consultants do pretty well because the limited K12 budgets mean that they usually need a guy who can walk in the door and do everything from switch configuration to spyware removal.
So my whole industry exists without having one person who only an Oracle DBA or Linux SysAdmin.
I live in the Midwest and almost everyone on our team (4 sysadmins) is in the 68-80k range. Around here that is good money. Consultants can pull in over six figures.
So I would say for certain that decent money can be made with a general skillset in public sector education.