I have both the a6000 (firmware 3.10) and a6500 (firmware 1.03).
When in Shutter Priority and identical scene conditions:
a6000 prefers to use f/4 which gets the subject in more focus which helps with a moving subject (the kids). Even if the subject has moved a little since the focus was acquired (which can happen even at 1/1000s with Continuous AF), the depth of field is wider (than at f/2) and the image would still be in some focus. Plus most lenses are sharper at f/4.
example: a6000 prefers to use f/4 and ISO 400 in my current lighting setting.
a6500, on the other hand, prefers to use f/2 which gets the subject in less focus plus most lenses are less sharper at f/2 than at f/4. (Sony lenses, most notably, are way not-sharper at f/2 than at f/4. The Sigma lenses keep the great sharpness all the way to their maximum aperture.)
example: a6500 prefers to use f/2 and ISO 100 in the same conditions in which a6000 prefers to use f/4 and ISO 400.
The noise difference between ISO 100 and ISO 400 is negligible, but the focus depth difference is big between f/4 and f/2.
In the P and P* (Program) modes, a6500 does prioritize the higher f-number for a wider focus depth instead of a lower ISO.
This happens with both Sony and Sigma lenses (like Sony Zeiss 24mm f/1.8 ZA or Sigma 30mm f/1.4).
I have reached out to Sony Chat Support and they have basically confirmed this change in behavior between a6000 and a6500.
Their response was to use Manual Mode.
While that is a solution, it's far from ideal as I would have to worry about reaching the limits of the ISO range and manually adjusting the Aperture. Otherwise, the photo will be under or over exposed.
That's something the a6000 takes care well by itself in the Shutter Priority mode. The camera increases or decreases the aperture accordingly to get a proper exposure when the limits of the ISO range are reached. All this while trying to use the f/4 aperture which gets the sharper and the more-in-focus photos.
Using the P Program Mode is also not ideal since it resets to 1/60s every time I power the camera on which is totally not suited for taking photos of the kids.
The IBIS of the a6500 helps with the static subjects at 1/60s, but not with the running around of the kids.
I have asked Sony Chat Support to inform the Sony a6500 product team to add this configuration option in their next firmware.
But I can't hold my breath until that happens, if ever.
On a personal note, this is a deal breaker for me in using the a6500 and not returning it to the store.
I need the camera to start with a specific shutter value when powered on and to favor the f/4 aperture which gets sharp and in-focus photos.
That's why I got the Sigma f/1.4 to still be able to have the fast shutter speed when coming home from work and seeing the wonders the kids do.
Question #1: I want to configure a6500 to behave like a6000. How do I do this? Is there a way to have this that Sony Chat Support hasn't thought of?
Question #2: Does this happen with your a6500 or a6300 as well?