Maybe one day when I get too old to just go skiing, I will need a training regime.
However, if you do nothing all summer and then suddenly spend a full day doing high-g turns, you won't be able to do your best the following day. Your body needs to get used to it. Ski vacations are expensive, and you don't want to spend them on "rest" days. When I was young it didn't matter, but I was in great shape. Now I find it's better to work up to it by spending a few afternoons skiing before spending a full day, though sometimes I do start out full day full speed, and rest the next day; I don't think I would want to have to rest on a ski vacation though.
As to biking as training, as has been said, it's not enough as heavy bike training does weird things to your body, like lowering your bone density. Your body adapts to what it has to do. Skiing is the best training for skiing. If you need extra strength and endurance to withstand that last compression at the end of the DH, go do some squats, but I prefer skiing through some compressions.
As a training for good conditioning, you need to add a weight bearing exercise. A little weight lifting, or even just using your body weight creatively in exercises would be a good complement to biking or swimming.
Training is required for world-level competition, not for recreational skiing. If you want to be competitive with the guys on the WC, you will have to train; if you just want to have fun skiing, forgetaboutit.