I am going to spend money I don't really have...
How much of a budget
don't you really have? While the suggestion of a 300/2.8L IS is wonderful, I'm going to assume that spending several thousand dollars is not an option...
Short version: I'd recommend the 100-400mm - the current one. IQ is slightly better than the 70-200 II + 2x, the usability is much better, and it's much kinder on the wallet.
Long version:
I personally own both the 100-400mm and the 70-200mm f/2.8
L IS II, plus the 1.4x and 2x II extenders, and both FF and APS-C bodies - so, I'm speaking from personal experience here.
Looking at your other suggestions
- new 100-400/4-5.6L - lenses that are rumors don't take good pictures. As pointed out above, Canon's recent track record for delivering even products that have been announced is pretty poor lately. Sure, if it's available when you're ready to buy, and you can afford it (the $3K price estimate seems quite likely to me), get it.
- 70-300L - a very good lens, and if you don't need 400mm, it's the best choice. But, in my experience, for shooting birds/wildlife, you often do need the reach of 400mm on APS-C, and you may end up cropping even more.
- 70-200/4L+2x - a non-starter. You'd be at f/8, meaning no AF unless you also buy a 1-series body - except the 1D X - to mount it on.
- 70-200/2.8L+2x - this one is worthy of more discussion, and it seems you think so, too, based on your post title
converters: so far, I've only read excellent reviews about the Canon 1.4x/2x mkiii - they sacrifice some af speed, but iq is said to be excellent. Am I missing something here?
The extenders are really designed for use with supertele primes (300/2.8 and up, excluding the 300/4 and 400/5.6), which take much less of an IQ hit than any zoom or the cheaper primes like the 300/4). Canon knows this - it's why they designed the new 200-400mm with an integral extender, one that could be optimized for that zoom lens, even though there is already an excellent 1.4x III. IMO, an extender should be thought of as an occasional-use item. In practice, mounting the extender is a pain (unless you happen to have three hands), especially in the field and if you're in a hurry. I don't think it's wise to buy a lens and plan to use it mostly with an extender. Rather, get the native focal length you need, and use the extender as a supplement once in a while, for specific reasons.
In this specific case, the first question is, which 70-200/2.8 do you have in mind - there are three of them. You mention that image stabilization is important, so that rules out the 70-200/2.8 non-IS. I would not use a 2x extender on the 70-200/2.8 IS MkI, it takes a big IQ hit - here's a
comparison. The 70-200mm f/2.8L IS II does take extenders reasonably well - here's a
comparison of bare MkII lens vs. lens+2xIII. Still, considering just IQ, the 100-400mm at 400mm is still better than the 70-200 II + 2xIII (
comparison).
In practice, the IQ difference between the 100-400mm and the 70-200 II + 2x is probably not going to be too noticeable (or not at all) in real-world shooting. But also, as you mention, using an extender slows down AF - in fact, with the 2x extender there's a
50% reduction in AF speed. That means the 100-400mm will focus faster than the 70-200 II + 2xIII (it's noticable in real-world use, but it's not
too bad). Finally, there's cost - the 100-400mm is over $1000 less than the 70-200 II + 2xIII.
You mention the variable aperture as a problem, but in Av mode it really isn't. It can be annoying if you shoot full manual, yes...but depending on your body, Auto ISO may compensate (on the 7D, M mode supports Auto ISO, which would compensate for the variable aperture as you zoom).
You mention using the lens for close-ups, and in that situation, the 70-200 II + 2x has an advantage - the bare lenses have about the same maximum magnification (~0.2x), but if you add the 2x extender to the 70-200 II, you double the magnification without changing the MFD, meaning you can achieve 0.4x with the combo. However, keep in mind that applies at the MFD, which is 4 feet for the 70-200 II and 6 feet for the 100-400mm. So, unless you can get pretty close to your subjects, the extra magnification won't necessarily be double.
So...comparing the current 100-400mm to the 70-200 II + 2x, which is better? Well, the 100-400mm has slightly better IQ at 400mm, slightly faster AF, and is substantially cheaper. The trade off is the 100-400mm is an older design (not sure why that matters), has a lower maximum magnification, and is not weather sealed (which really only matters if your crop body is the 7D).
The other problem with the 70-200 + 2x is usability. In one sense, it's more versatile - you have a fast 70-200mm, and a slow 280-400mm, all in one lens. But you don't have both at once, so, it's often an either-or situation anyway.
Personally, if I know I'm going out to shoot birds/wildlife, I grab the 7D and 100-400mm, almost every time. The only exception to that is when I'm going out in the rain - in that case, I'll take the 7D and 70-200 II + 2x for the weather sealing. Honestly, I found the 70-200mm focal length to be a bit awkward on APS-C - too long for indoor use, too short when I needed real reach (since I have the 100-400 for that). OTOH, on FF the 70-200mm is an amazingly versatile lens, great indoors and out - so, if I'm going on an outing and taking the 70-200mm for that, I'll often throw in the 2x extender, just in case.
So, the bottom line is that for birds/wildlife/etc., I think the 100-400mm offers the best compromises between IQ, ease of use, and cost. I don't think there's a better option under $3000 if you need 400mm and want IS.
I'll end with a bunch of examples, since I think it's helpful to back up words with images. You can click through for a larger image, and View All Sizes in the upper right can get you a 1600 pixel version so you can get a better idea of the IQ.
First, 70-200 II + 1.4x, on a drizzly afternoon:

EOS 7D, EF 70-200mm f/2.8
L II IS USM + EF 1.4x II Extender @ 280mm, 1/2000 s, f/6.3, ISO 3200
Next, 70-200 II + 2x, again on a misty day:

EOS 7D, EF 70-200mm f/2.8
L II IS USM + EF 2x II Extender @ 400mm, 1/160 s, f/5.6, ISO 3200
Now, a few with the 100-400mm:

EOS 7D, EF 100-400mm f/4.5-5.6
L IS USM @ 400mm, 1/640 s, f/5.6, ISO 125

EOS 7D, EF 100-400mm f/4.5-5.6
L IS USM @ 400mm, 1/640 s, f/5.6, ISO 200

EOS 7D, EF 100-400mm f/4.5-5.6
L IS USM @ 400mm, 1/640 s, f/5.6, ISO 160

EOS 7D, EF 100-400mm f/4.5-5.6
L IS USM @ 400mm, 1/1600 s, f/6.3, ISO 1600
Finally, since you mention shooting insects/plants, too, here are a couple of other subjects with the 100-400mm:

EOS 7D, EF 100-400mm f/4.5-5.6
L IS USM @ 400mm, 1/640 s, f/5.6, ISO 100

EOS 7D, EF 100-400mm f/4.5-5.6
L IS USM @ 400mm, 1/640 s, f/5.6, ISO 100