So I've been staring at my D5300 and realizing the kit lens just isn't cutting it for the distance shots I want to take at the local zoo next month. I'm trying to stay under $250 which is kind of a tight budget but I've been doing some reading and now I'm just more confused.

I keep seeing the AF-P DX 70-300mm f/4.5-6.3G ED VR pop up in every list of cheap zooms but then some old forum posts say the AF-P lenses wont work on older bodies or they lack a physical VR switch which sounds super annoying if I want to use a tripod. My logic was that the extra reach is good but then I saw the 55-200mm VR II. It's way smaller and maybe better for carrying around all day but is the reach enough for animals that are further away?

I'm getting really stuck on the compatibility thing too because I've read that the AF-P focus might not even work on some older DX models without a firmware update and I'm not sure I want to mess with that. Is the image quality on that 70-300mm actually that much better than the old 55-200mm or am I just overthinking the tech specs again? I just want something sharp enough that my kids dont look like blurry blobs when they're across the park...