Photography

The words “low light”, “400mm”, “wildlife” (implying excellent autofocus), “DSLR” (old), and especially “budget” don’t belong together.

Well there is a 800mm f5.6, and a 600mm f4, and a 400mm f2.8, but of course they are bulky and pricey. More light needs bigger glass.

And then there's budget lenses. You can't have it all at once.

Cause physics. F-number is calculated by (focus length / diameter of entrance pupil). Say you want a 400mm focal length with max f2.8, your front element diameter would need to be 400/2.8 = 143mm. Looking at the Canon EF 200mm F2.8, which has a filter diameter of 72mm, 200/2.8 = 72. Math checks out.
(Edit: Front element diameter is not actually equal to entrance pupil diameter, but here let's just say it is. You can look up the physics if you're interested.)

Have you actually looked up the lenses yourself? Canon EF 400mm f2.8 lens exist, but you're looking at 10x your budget brand new, and really huge sizes. And if you want zoom on top of that? Just search the Sigma 200-500mm f/2.8, aka the "Bigma". I'd say you're better off using f4 lenses or whatever is within your budget, and try to work within the limits.

TLDR: If you want telephoto with super wide aperture, the lens would need to be massive, and expensive.

Unfortunately, you're asking for the worst-case scenario for lenses. This has to do with physics, and how much glass is required to shape the light onto the camera sensor. These lenses exist.. but the prices are ooof