I might be able to get a Sigma 60-600mm DG HSM (not DN) lens for a good price, used, which has an A mount. I have an E mount body (a7r3), and was wondering if, combined with an A to E adapter (probably this one?), it would work reasonably similarly to the DG DN version of the same lens, since they're both full-frame lenses. Or does the adapter cause a loss of aperture or cause autofocus or other issues?
There can be some downsides to using these LA-EA* adapters with Sony E-mount cameras, compared to native lenses, particularly with previous generation cameras/adapters. Sony publishes a list of restrictions on a lens-by-lens basis, but only for Sony/Minolta lenses. I would not assume that a third-party lens will work any better than a Sony A-mount lens.
The latest adapter – LA-EA5 – should not cause any loss of light transmission. This adapter has no translucent mirror to deflect light to an autofocus sensor like some earlier adapters; rather, autofocus is handled entirely by the host camera's sensor. Some autofocus modes/features may not work fully (e.g. eye-detection) – these are the kinds of restrictions to watch out for. Watch out for limitations in video mode too.
Sony publishes an online Help Guide for each camera model. Unfortunately the guide for the α7RIII does not cover the LA-EA5, but take a look at the guide for the α7RIV where some of the restrictions for this combo are mentioned.
Note of course that some of these restrictions may not be deal-breakers. At the end of the day, it's the image that counts.
What matters are downsides that matter to you and how those compare to the upsides that matter to you.
For example adapters add weight and bulk and are a potential point of failure. But a 60-600mm lens already has weight and bulk so you might not care about the weight and bulk an adapter brings.
Or you might. How others would choose doesn’t change that. I mean some people choose large format film despite the downsides compared with digital mirrorless because what matters to them even though most people choose differently.
In the end what matters are the pictures. A 60-600mm lens is likely to produce pictures that wouldn’t happen otherwise.