A 70-300mm is "not that long". If the wildlife is large, like say...elk or bison which are herd animals in some national parks, and the animals allow people to get within 40 yards of them, then yeah, a 70-300 zoom is okay. If by wildlife you mean ducks and geese at 80 yards on a duck pond...a 70-300 is more of an "overview" lens, and will not allow you to select individual birds unless you can get CLOSE.
The problem is that long lenses that are really "good" for wildlife are expensive. The "best value" in a normal, everyday tele-zoom is indeed usually the camera maker's own 70-300mm lens. Next is the camera maker's 70-400 or 80-400 lens. As I understand it, the Sony-branded lens in that category is very good.
The hot, new thing is the brand-new Tamron 150mm-600mm f/5~f/6.3 zoom with built-in image stabilizer in Canon and Nikon mounts. This lens is right around $1,000, and it offers a lot of reach for the $1,000 price. Not sure when it will be released in SONY Alpha mount; the released Canon first, and then it took 30 to 60 days before the Nikon F mount version was released; I suspect SONY mount will be next.
ALL of the above lenses, from the 70-300 to the 150-600mm are pretty much designed for use in GOOD weather, meaning bright light levels.