Not necessarily. A 50 mm f/1.8 and a little cropping would be fine, at least to start for a moderate price
You have a good eye and a solid style, and that's the hard part. A wide aperture to blur the fore/background naturally would help with that 50/1.8 lens. Look into making a DIY lightbox to get some softer, more controlled light with softer shadows and fewer harsh reflections
Lighting is definitely hard, but technically it should be a little easier at the scale of board games. The geometry works out such that cheaper and smaller softboxes, umbrellas, reflectors, and other equipment will be easier to buy and easier to control than much larger gear for larger subjects. A small umbrella over a tiny board game piece is effectively enormous (good). Look into a full video course on studio lighting, and flash photography over on Lynda or Skillshare - it'll be like an hour or two long in total. If you go to college, they may hook you up with free Lynda account like FSU does for students and alumni. Your local library may also give you access.
I'm most comfortable with Canon because I have one, and there seems to be a wider selection of lenses, but I think you're fine here. That's a great camera and Sonys are known to work really well indoors in low light. Definitely not a misstep, but there's a lot more to photography than just buying a nice camera. Check out YouTube for LOTS of tutorials on post processing.
P.S. Lightroom/Photoshop are the industry leaders in post processing, but I am cheap and I like RawTherapee/Gimp which are the free, open source alternatives. Editing is just as important. Take the skills you learned on your iPhone editing software and continue doing the same kinds of things on desktop (or mobile) but with a bit more control. Good luck.