Below are some long, sometimes technical answers. The short answers are:
- if you really want to shot tiny birds at long distances (songbirds at 25 - 50 yards), you need the P1000 or something else with a 2000 - 3000 mm equivalent lens. If that's not the case and you're working more like 10 yards or less, than the d7500 + 150-600 is useful.
- In low light, I'm still going to say the DSLR will win over a bridge camera. You'll get higher ISO and better dynamic range, such as something in a ray of sunlight preserving shadow details or the micro contrast on feathers.
- Winning in any one category, of course, doesn't mean it's right for you. I still suggest you try before you buy.
Note: I think in the past I said the P1000 is 2000mm equivalent, but it's really 3000mm equivalent.
I'm assuming by songbird, you mean something under 10" and more like 5" - 7". Is that wrong? Also, in this answer, I'm assuming we're more-or-less in the linear region of the tangent function, so I blithely use linear scaling. I'm ballparking some of these numbers, so take them for order-of-magnitude not precise calculations. Also, I'm assuming you want the resulting photo to be at least 5MP - 10MP, not some super tiny thing.
To answer your question: a DSLR will have a hard time at this (maybe a d850 or a7riii with 600mm lens plus heavy cropping will get close). The P1000 would be the choice here.
For some perspective, I put a 7"x5" postcard on a tree. With a 500mm lens on full frame, the postcard fills the image at about 9.2 feet (say 3 yards). On a DX crop sensor, it's about 15 feet (say 5 yards). Those are both as-measured on a d850 in FX and DX modes. A 3000mm lens (6x distance) would only be 18 yards (that's approximate).
A d7500 with a 600mm (900mm equivalent) lens would be at about 6 yards. You would need a 4x crop in editing which would bring the 21MP sensor down to 1.3MP (it's square because MP is area). That's not workable. A more practical size would be a 2x - 3x crop, so the songbird would be about 1/3 - 1/2 of the frame width.
I've shot the d850 with 600mm and cropped down from 45MP to about 5MP. That gives me a 2000mm equivalent focal length. Doing that, the song bird would be a bit under 1/2 the width of the cropped image.
At 25 yards, a P1000 would need a 1.4x extra crop, which it's 16MP sensor might deliver (especially editing down from a raw file).
If you're into math, a 200mm lens has a 10.3* horizontal angle of view. From trig, if the width of a subject is w, then your distance d = w / (2 * tan(a/2)). For w = 0.6ft (~7") and a=10.3*, d = 3.3ft. For 500mm, a = 10.3/2.5 = 4.12*, so d= 8.3 ft (compared to my measured 9.2 ft -- the actual angle of view of the lenses vary a bit). In any case, if you have d = 75ft, w = 0.6ft, then a= 0.46*, so you would need about a 4400 mm equivalent focal length.
Some other rants about IQ. First, you need good shooting discipline (technique) to get top IQ. Otherwise you'll get blur from bad settings, camera shake, mirror bounce, etc. Generally, the larger the sensor the better the low-light performance and the dynamic range. We've not talked DR yet. It's how well a camera can render shadows and highlights. A high DR means you maintain details in deeper shadows and brighter highlights than a lower DR. There's also the lens quality (often seen as MTF charts) -- how much distortion and aberration it has. Some cameras correct distortion in the body, but that can lead to lower IQ because it can blur fine details. So, there are lots of fine details that affect IQ.
Event worse than above. When I take photos of songbirds, it's under 10 yards and with an equivalent 1200mm - 1500mm lens (including cropping in post).
Full frame, low megapixel DSLR. Something like the d750 or d5 or Z6 or d500 (or equivalent Canons), or Sony a7iii or a7sII or a9. Not listed in any particular order.
In something in your price range, probably the d7500 (or equivalent Canon). Or Sony a6500.
The longer the distance, the longer the needed equivalent focal length. Equivalent focal length is the real lens focal length times the sensor crop factor times the edited crop factor. A DX sensor is a 1.5x crop factor (A full frame sensor is 1.5x bigger [usually measures along the diagonal] than a DX sensor). Micro 4/3 is a 2x crop factor. The P1000 is 5.6x crop factor. The editing crop factor is the amount of extra you cut off in your editing software.
In general, the larger the sensor crop factor, the worse the low light performance, for equivalent megapixels. In general, you can crop down to about 5MP - 10 MP. Let's just say 10MP. A 20 MP camera could crop down 1.4x to get to 10 MP (it's square root because MP is an area measurement). A 16 MP sensor (say P1000) would be 1.3x crop factor down to 10 MP. In my experience, the P900 did not crop well at least from the jpeg and did not have a raw option. The P1000 might have better jpeg and has a raw option, so you might be able to get that 1.3x crop.
There's a few aspects to answering this.
1) A 7500 + 150-600 is about 6lb combined. A p1000 is about 3 lb. I use the d850 + 150-600 (maybe 7lb combined). I have no problem hand holding it for quick shots or BIF. But, I often want to wait for a perched bird to do something interesting. I could not hold 6lb - 8lb on target ready to shoot for more than a half a minute. I use a monopod and that lets me keep aim on a target and ready to shutter release for essentially as long as my feet hold up. But clearly the P1000 wins here.
2) Shooting at 500mm equivalent or above becomes hard to find things and track things (it gets easier with practice). Shooting on a monopod or a gimbal head tripod makes it a lot easier. The P900 has a feature to press a button to zoom out and when you release it goes back to your original zoom. This lets you find something then zoom in. i assume the P1000 has the same feature. This can really help hand holding. In any case, shooting at really high equivalent focal lengths needs steady hands and a support really helps.
3) you need a good way to carry the load. The P1000 is not too bad, but for any of these 6lb or above combos I would use the black rapids sling strap.
Yes. Your main options are Nikon, Canon, or Sony.
We've not talked about Sony, but they are a solid platform too and have some impressive high ISO performance. But they can get expensive and there's no good long lens options in your price range. You can do something like an a6500/a6400 + tamron/sig 150-600 + mount adapter, but I've seen mixed things about the AF performance and I've not tried that combo.
I don't think it's "the way the industry is going" so much as a way to sell the tiny sensors in a package that fills a niche that full frame and APS-C sensors do not (either DSLR or mirrorless). You can get either tiny cameras with normal zoom ranges or small(ish) cameras with insane zoom ranges.