Many mirrorless camera systems are available these days.
What are your expectations out of this "point and shoot"???????
If you want to have a comparable camera body in Canon 7d (mk2) AF speed I don't suggest the fuji bodies unless it's the newer body like the X-t20, X-E3 or pony up to the X-T2 or X-pro2. However the Sony offerings is probably better in AF-C faster focusing mirrorless bodies compared to Fuji. The Fuji X100 series is a fixed lens and I'd prefer an interchangeable lens system for "point/shoot" applications.
You can go the Micro 4/3 route but price wise they are pretty expensive now with their newer generation bodies. Unlike in the early years the price point was much better. However as you know your 7D mk1 will be easily outperformed in image quality with the newer smaller M43 systems. Pricing out the m43 lenses is not inexpensive and the system still is somewhat crippled in high iso and dynamic range capabilities. However if it suites your needs the system may just fit.
I've played with Sony systems but for slower moving subjects (not really for sports / fast action application) the Fuji system is quite nice. The X-t10 16mp camera (similar AF system and same sensor to the older flagship X-t1) paired with the the Fuji lenses is amazing even though it's a slightly older model. The crop sensor Fuji produces extremely nice files with fuji flavour. Even the 18-55 kit lens is f/2.8-4.0 variable aperture with excellent pro build but the IQ is remarkable for a kit lens.
Shop around and play with the ergonomics of each system. Have fun spending money