If in the market for a 500-600ish lens, consider the lens size/weight factor.
Been where you currently are and not always amenable when kids in tow and they want to scoot off here there and everywhere.
(my side issue is a bung knee too tho).

But Nikons 200-500mm is very good, from what I've seen the Sigma(Contemporary) and the Tamron 150-600's are also both good lenses.

Sigma Sport 150-600 is also very nice(I have) and so far with the limited use I've had out of it .. I like it very much.
Does feel very weird on the D5500 tho .. far too big and out of place(hand held), on a tripod tho make no difference.

Interesting that you're using the back button AF method! From my reading up, I didn't think that the D3500(my D5500 doesn't) have that feature. So if you do have it, I assume you use the AE-L/AF-L button, I suggest you stick with it.
Although it's easy to say with the larger bodied cameras like the D300 D800 and so on, I dunno how well or not it works on the smaller bodied cameras, in terms of ergonomics.

Have to go check the D5500 now to see if it has that feature too.

Also, technically it's called the "AF-On" focus method. Some cameras have a dedicated AF-On button for that purpose (I won't get one that doesn't have it foir my personal use). I use it all the time for every situation.
It allows you to control the AF a little better, I feel it's more ergonomic using the rear button. Keep the shutter half pressed all the while(for exposure).

Remember here too tho. You have consumer gear .. camera and lens. Don't expect professional level performance out of it. That could mean that AF can sometimes be a little hit and miss .. pretty much normal.
What will be satisfying for you tho is that when you do get really nice shots(and your later thread shows that) .. it feels more personally satisfying that they were shot with consumer level gear!

Now, if you just use [A] mode and retain control of aperture, you decide if you want f/6.3, f/7.1 or f/8. You may even want f/9 sometimes!! You may even want to drop down to f/5.6 .. zoom out to about 220mm on the lens .. instant f/5.6
But keep that AutoISO in mind, set the min shutter, use that auto setting set to one notch to the left(slower). So that at 300mm, you should get roughly 1/250 or so shutter, but if you zoom out, you'll get slower again, say 1/180 at 200mm.
Remember thats the idea behind auto shutter speed.
See how that works for 'ya for a bit, it may or may not.

But note that if you demand higher shutter speeds and dont' have enough aperture to allow these, then all you end up with is under exposed images, and at high ISO.
This is where noise will become more obvious, and why many folks say that "high ISO is bad", or "keep it low" .. etc.
In my mucking abouts with this sensor, I've never seen problems at ISO6400, as long as shots have been exposed well. Even with a slight crop.

Two images, relatively high ISO, not quite nice lighting(fluoro)
Both from the same D5500(same sensor as your D3500)
D5500_DSC_0183.JPG

D5500_DSC_0183_01.JPG

Guess which one is higher ISO. Try to guess ISO value, and also which one has had NR applied?
Both images converted in Nikon's (free) software, to which, other than NR applied to one image, no other editing done.
But both are uncropped.
Unfortunately the Nikon 105VR(lens used) isnt' the sharpest tool in the shed either .. which reminds me .. gots to get me a new Macro lens sooner rather than later. And it hasn't misfocused so much as it's a DOF issue in these images.
The DOF is so shallow, the focus is on the skin, with some dust/hair/fur detail, but the seedy looking things are recessed into the skin so OOF due to the thin DOF.