Birding is tough on a budget.
The "cheap" lenses that birders love are
1) the Sigma 150-600mm in the "contemporary" version (Sigma makes the same lens in the "sport" version and it's a better lens ... but it's literally about twice the price. The "contemporary" is just under $900. The "sport" is just under $1800.
2) the Tamron 150-600mm ... note that there's the original which is a little less than $900 (and just a few dollars less than the Sigma) but Tamron also somewhat recently released the "G2" version which I think runs around $1400.
And these are the "cheap" lenses (because the lenses they really prefer are $2000 and up. So the definition of "cheap" is all relative.)
Do you have to use these lenses? Certainly not. Birding isn't easy... but these lenses make it a little easier. If you're trying to capture birds in flight, you need a lens that is responsive - snappy auto-focus and capable of auto-aperture changes. If you can find a non-moving bird, things are easier. But many birds are shy and it's difficult to get close enough for a well-framed shot before they fly off.
Lenses like those 500mm f/8 lenses are mirror lenses (catadioptric lenses). There are no aperture blades so exposure can only be adjusted via shutter speed (or ISO). Out-of focus blur is donut-shaped because the lens has a secondary mirror which creates a central obstruction. They are not auto-focusing lenses.
Trying to capture a shy create or a bird in flight with such a lens will be... quite a challenge (to put it mildly).
Extenders (focal length multipliers) cannot be used with just any lens. Canon has a specific list of lenses that are compatible with their 1.4x and 2.x extenders. They not only alter the focal length of the lens, they also affect the focal ratio of the lens.
If you use the 75-300mm lens (at the 300mm end - which is basically where you'll be) the lowest focal ratio is f/5.6. When you use a 1.4x extender you multiply BOTH the focal length AND the focal ratio by 1.4. This gives you an effective 420mm f/8 lens. Your camera will NOT have working auto-focus at f/8 (only a few cameras can auto-focus at f/8 and they're all high-end models.) Even so... auto-focus speed is reduced with the extender. And a 2x adapter doubles things... so the lens effectively becomes a 600mm f/11 lens (and there are zero cameras on the market that have working auto-focus at f/11).
You also take a hit in image quality when you use a focal length multiplier. Canon's latest version "III" extenders are extremely good. Third party extenders like the Kenko brand will result in quite a bit of a noticeable hit in image quality. But that assumes the lens is even compatible with a focal length multiplier AND that the camera can still have working auto-focus at the new max aperture value available with the multiplier on the lens.
You may be able to stretch the budget by looking for used lenses (hopefully in good shape).
The tele-extender isn't going to work for the 75-300mm lens. The mirror-lenses you mentioned will "work" but probably frustrate you. With that said... there's a nearby nature park that is a natural aviary for blue herons and sandhill cranes. Sandhill cranes are normally so shy that if if you spot them on the far end of a large field, by the time you step onto the field, they're gone... they're not sticking around to see if you're planning to get within 100 yards of them. But at this particular aviary, they were shockingly used to humans. I watched a mated pair wandering around foraging for food with their chicks ... and the chicks were walking up to people ... (literally within 2-3' and in one case I saw a chick go right up to an adults shoes (I think he may have been eyeing the laces in hopes that they were worms.)) That's "not" typical behavior. I had to ditch my 300mm lens for a 70-200mm.
I think this was shot at 140mm (and I cropped in very gently to improve composition).
So it's certainly possible to do bird photography with any lens (the above shot was not exactly challenging -- it was the proverbial "fish in a barrel" situation). But for most bird photography, birders prefer the native 150-600mm zoom range and the cost of the entry-priced lenses in that category are around $900 (but that's retail price on a "new" lens. You may be able to find a well-mainted used lens for much less. Especially as owners sell off their old/original versions to buy new/improved versions.
Actually it is listed and shown as SIGMA 170-500mm F5-6.3 APO DG
See pic.
