Then, of course, I would have to purchase the Mach3 software.
So, my second question: Does that software stack even seem like it would work or am I completely off base?
I do not want to drop $1000 on a machine only to find out I need to spend a bunch more on software that I would need to learn before I could even start to use it.
1) You can use Mach3 in demo mode for free, for up to 500 lines of g-code. Most simple 2D parts would be much less than 500 lines of g-code.
If you do decide to pay for it, I highly recommend against it. Mach3 is obsolete, with development having stopped over 5 years ago. I'd recommend UCCNC. It's cheaper than Mach3, for both hardware and software. They are currently offering a free license with the purchase of their UC100 controller. UC100 USB motion controller + free UCCNC software license key - CNCdrive - webshop
2) I'm not a fan of Sketchup for CAD, but plenty of people use it. Fusion 360 is free, has both CAD and CAM, and is far more powerful.
There are also plenty of free CAM programs out there.
3) Be aware that $1000 is very little money for a CNC. It would be like buying as new table saw for $40. A decent quality 4'x4' machine would normally be in the $4000-$6000 range.
As David says, your expectations will dictate what you think of the machine.
The machine will definitely cut your parts, but you'll need to make multiple, light passes. $1000 is small price to pay to get a feel for it. If you like it, chances are you'll upgrade to something better.