The key fob "authenticates" the car and allows the driver to shift into gear for up to two minutes after a key command has been sent to the car by the fob. So if you unlock the car with the fob and wait longer than two minutes, then get in, it will ask you to authenticate again by placing the fob or key card behind the cupholders.
As you mention in your post, you can avoid digging the fob out of your pocket when prompted for this message by double clicking the fob in your pocket to send the unlock signal and re-authenticate the car again.
There is no way to disable this. The Model 3 does NOT have passive entry and cannot detect the fob when it's in the car. It is similar to a traditional key fob with the exception of its NFC capabilities should the battery die, but it does not actively emit a signal (as other keyless fobs do) that the car can detect when you're in close proximity with it.
If you're unlocking the car and immediately entering and trying to shift into drive/reverse and it's asking you to authenticate then, there is a problem as this is not the expected behavior. But if you're leaving your Model 3 unlocked and simply getting into it without pressing any buttons on the fob, then no, it's not going to allow you to change gears.