Thankfully there do exist alarm clocks that specifically cater to this problem. Philips had an alarm clock that would light up slowly (using a halogen bulb) and optionally play the radio or (oddly enough) the sounds of cows in an Alpine meadow, or a Buddhist gong, or birds chirping (the radio turned out vastly preferable).

The clock display was backlit by orange LEDs and seemed even less noticeable than the classic red LED display you mention.