I think it depends on what kind of sports you're shooting, and under what conditions.
I once had a 70-300 IS USM (the non-L to which you refer) & used it extensively. It was OK but on the long end anything over 200mm was really not so hot. Photozone
& other tests confirm this. I was using it on a 40D and your 1DsII has a much lower pixel density so it won't push the resolution limitations of this lens as hard, but I'm still not sure this lens is such a good choice for you. Also the AF is not full "ring USM" so you don't have full-time focus adjust available & the AF is not as fast as a full "ring USM" lens. In short, it's OK but not great.
If you're shooting sports where you can get a bit closer to the action (e.g. indoor sports or something outdoors where you're not too far away), you may want to consider a used or refurb 70-200 F/4 L. I started out with one of these & it was really great. Super sharp at all focal lengths, fast AF, not too heavy. I sold it because I found myself using my 135L most of the time instead, and was quite long & didn't fit well in my bags or balance so well with my smaller bodies. But with a 1Ds you're already used to dragging around bigger gear, so this shouldn't be a problem for you. When refurbs & used lenses are in stock they're under $500
so just keep an eye on it (also check B&H & Adorama used inventory).
Also all the 70-200s (unlike the various 70-300s) work great with the 1.4x and 2x teleconverters. The mkII 1.4x is quite good but if you were going to go for a 2x TC I'd say either splurge for the mk3 or don't mess with it. But a used/refurb 1.4x II turns the 70-200 f/4 into a 100-280 f/5.6 with very good quality & fast AF for not a lot of money. Removing a TC takes more time than just turning a zoom ring, but the benefit is that your base lens is lighter and more luminous for closer indoor stuff.
You only need to worry about IS if you're shooting in situations where you can't use a tripod. For a shorter lighter lens like the f/4 zooms a monopod is also quite acceptable. IMHO a tripod collar is necessary for the heavier lenses, (use a 3rd party tripod collar or find a used Canon collar as the new ones are ultra-pricey) but especailly on a bigger body like the 1DsII the f/4 zooms should be fine using the camera's tripod mount & a good ball head. For sports, most likely you'll have your shutter speed fast enough to be able to hand-hold a 200mm lens anyway in order to freeze the action. If you've got a monopod, you can really move around quickly & easily & still get down to 1/100th or so at 200mm with good shooting technique. Your subject is probably going to be pretty blurry at that speed anyway unless they're standing still.
The Tamron 70-300 VC is supposed to be the nicest of the budget zooms, but unless you really need the IS I'd stick with the white ones with the red rings :-).