Hopefully you're talking stationary birds as 6FPS makes shooting moving birds really tough. Shooting at 40 fps I can tell you that often I still don't get a good frame in a one-second burst or just a few in the pile. I'm not as good as many on this site are, so if you are better than me, then perhaps you can succeed at 6fps. IS and IBIS really help on the low-light issue. I cannot believes some of the sharp 800mm hand-held shots I've gotten using having both IBIS and IS.
The lowest cost IBIS solution will likely be a used or refurbished R7s when the R7 Mark ii gets announced, though used R7s are so plentiful their used prices are getting close to the R10. You can still get 23 fps for one-second bursts on the R10 (vs the approx 1.5 second buffer on the R7), but you give up IBIS. On a Canon RF APS-C I would say the Canon RF 100-400mm f/5.6-8 IS USM lens would be the best lens value for birds. The Sigma 16-300 is a cool lens, but the autofocus is less robust and AF speed and tracking is really important for birds.
I don't know what your budget is but shooting birds is pretty tough work and specialized gear is a huge plus. Even used (and there are a lot of used R7s in the market right now - I don't like the control layout but I'm sure I could get used to it) the R7 camera runs about $1,200 and the 100-400mm is $700 used and $700 new - no brainer on that one to get one new. So you're looking at about 2,000 $US for that combo.
If you're heavily invested into your existing camera the Canon EF 100-400mm f/4.5-5.6L IS II USM is probably the way to go. The Mark i version of that lens has half the IS and slower AF and the Mark ii has internal zoom rather than push-pull. It is a great birding lens for your current camera if you're just wanting to get closer to mostly still subjects. About $1,800 used and $2,700 new.
Personally I recommend considering taking the leap to mirrorless APS-C and the RF lineup to get the faster frame rate and IBIS on the R7, but you can get some great results with the EF 100-400 glass, certainly.
Refurbished from Canon is always a great deal if you can get lucky and get one.
>> Owns/Owned both Canon EOS mirrorless full-frame and APS-C cameras and associated RF, RF-S and EF adapted lenses - inventory tends to change on short notice. Same for flashes, tripods, bags, straps, etc.
Plus>> Canon imagePROGRAF PRO-1100 Printer
>>The opinions and assistance are my own. Please don't blame Canon for any mistakes on my part.
