Some of these comments confuse me some.
For your needs you are totally on the right track IMO.
Lack of sharpness in a Sigma 30mm? Maybe to an upper end professional, but I guarantee you will never have that complaint taking family pics.
No need to go to a full frame for quality pictures. Full frame has advantages sure, but many of those advantages simply won't be noticeable in your hands.
Personally I would recommend skipping the kit lense. If you bought an A6000 with that kit lense and the Sigma 30mm I'd bet you'd find yourself disappointed with the kit lense and not using it much. Instead I'd recommend a combo of a body (I really like the a6300 for the price) and a combo of say that Sigma 30 and the Sigma 16 or maybe the 50 if it fit your needs better, could be had in your budget and cover an awful lot of your needs.
Bokeh isn't the be all end all of portraits, but its a lot of fun to play with, especially for portraits (and fish pics

) so the faster lense is nice. Ya you won't need to shoot wide open every time, but its nice to have it when you want to. I shoot the Sigma 30 and 16 wide open all the time and sharpness is simply not an issue. (Again, I'm not selling pictures but I have ZERO complaints about sharpness even when wide open). The shallow depth of field is a lot of fun to play with and can really make your portraits pop.
I'm no professional photographer but I have a lot of experience with Sony mirrorless and feel you would not come away disappointed going that route. Also keep in mind the crop factor of the APSC lenses. That 30mm on that body produces a 45mm focal range. 1.5 crop factor and all that. So a combo of the 16mm and the 30mm Sigma lenses gives you 24mm and 45mm focal range. There's a lot you can do with that combo. Later, if you decided, you could add a Sony 18-105 or the Sony16-55 for more versatility. There's several full frame options in the 24-70mm range that are great options as well. Again, with crop factor a 24-70 is more like 36-105mm so always keep that in mind.
Those Sigma lenses are flat out performers that really out perform their price class.
Sony has an enormous lense selection for their mirrorless bodies, and remember that you can use their full frame lenses on an APSC body no problem, which makes the lense options even larger. Many great third party lenses too.
There's just no reason that a mirrorless system wouldn't make you extremely happy given your experience level and the needs you listed.
Edited to add that a quick search shows the A6000 or A6300 as being harder to find than the last time I searched so you may have to consider used for the body if you decided to go that route. Both bodies have been discontinued but performance wise are just fine even today for your needs.