Yes, it would be more fun with bigger tires. I may still get 30 or 31 inch tires. I had my last 4x4 for five years and didn't do a lot of off-roading ,though that could change. I must admit, it was fun, but was always scared to do any real hard core off-roading for fear of rolling the vehicle or otherwise damaging it. Then again, it was a Lada Niva I used to have. Worked well when it was running OK, but just not the most reliable or durable vehicle. Plus I only had the stock tire size.
Every time I think I have my mind made up on tires, I end up reconsidering my decision. However, the stock tire size might be better for driving in snow, since it won't be as wide and can cut through the snow better. Then if I get into some serious off-roading, I can get another set of rims and tires (maybe mud tires) for summer or to put on when I want to go off-roading.
For the insurance, they want an appraisal, otherwise it's just insured for black book value. Not sure if they would frown upon any modifications, like a lift, 33" or bigger tires, etc.