I used to use the Sigma 100-300 f/4 HSM as my main baseball lens in the mid-2000's; it's got a good focal length range, but the 75-300mm range is also actually VERY useful for baseball, especially when you shoot from down by 3rd base. 3rd base and first base can be covered pretty well with short lenses!!! On Nikon's D2x with its built-in 1-button, INSTANT "Func." button switch to 2x crop/1.5x crop/2.0x crop action, the 100-300 was SWEET for outfield plays!!! (150mm-450mm FOV equiv at 1.5x crop at 5 frames per second, then 200mm-600mm FOV equivalent at 8.2 frames per second as soon as you pressed the button with your right middle fingertip!)
Baseball is one sport where you can PRE-focus on the bases, and be READY for the action, much,much more so than any other sport I can think of, at least on infield plays. And since it is played in daylight, in the spring and summer, the slower f/stops of the consumer lenses will be "okay". Also, the focusing is all at distances of 30 to 300 feet, so focusing "speed" is not a really huge issue in bright summertime light.MY best friend shoots for a minor-league team and has for seven years now. WHERE you sit affects HOW you shoot the games. Down by the dugout at 3rd base is a good, easy place to start shooting, as is the first base line.
The Sigma 100-300mm f/4 has Hypersonic Motor (aka HSM) focusing, a constant maximum aperture of f/4, and is hand-holdable and pretty good actually; I got mine used years ago for $500. Not sure what they sell for these days; it would be considered a step "up" from most of the 70-300mm lenses or 75-300mm lenses.