Why aren't departures separated by minimum distance rather than minimum time?
For the FAA(all my answers can be looked up, and expanded upon in FAA Order 7110.65W, Chapter 3 mainly, but I'll try to keep it short), most departures from the same runway use distance for separation. The standard is the lead aircraft is clear of the runway, but most of the time can be shorter under most circumstances. The only times it becomes time based, if it's wake turbulence related, or various conditions of non-radar separation needing to be applied.

In general for runways, aircraft are broken into 3 categories for separation(excluding weight class, which add wake turbulence time if necessary). Category I aircraft are light single engine propeller driven aircraft weighing less than 12,500 lbs (and helicopters). Category II aircraft are light twin engine propeller driven aircraft weighing less than 12,500 lbs. Category III is everything else.

I'll mainly discuss same runway departures, since there's many considerations and it gets complex, and the FAA Order 7110.65 does a fairly good job of depicting and showing this separation standard. For the same runway when both aircraft aircraft departing are category I, the min distance between them is 3,000 feet and airborne. This means that the first aircraft has to be 3,000 feet down the runway and airborne, before the next aircraft starts it's takeoff roll. If the trailing aircraft is a Category II and the first aircraft is a Category I or II,  the distance increases to a separation of 4,500 feet between aircraft. If either aircraft is a Category III aircraft, it's 6,000 feet and airborne required.

Now if both aircraft are IFR, there needs to be IFR separation insured after the runway separation. In most environments the minimum radar separation is 3 nautical miles between aircraft. Now, between jets, the runway separation of 6,000 feet and airborne, will often be right at 3nm between aircraft(due to the speed differential mainly). Other options include turning one aircraft off the path of the other by at least 15 degrees and have them diverge apart.

Wake turbulence is one of the few areas that time is applied. If a heavy(300,000 lbs or greater) is departing in front of anything else, the next aircraft can't start their departure roll for 2 minutes when both are departing from the same intersection on the runway, and 3 minutes if departing the runway from different intersections. And a Super(A380/AN225) I believe are 3 minutes, when departing the same intersection, but I've never worked them in the tower. There are a few other times when time is used, intersections, opposite direction.

For arrivals, all except non-radar separation is distance based.