I was reading some old NTSB statistics and found taildraggers have a much higher accident rate than planes equipped with tricycle landing gear.
Would a shock absorber incorporated into the suspension to absorb the landing shock and slow the rebound prevent bounce and reduce mishaps?
Taildragger (CLG) vs tricycle accident rates are not vastly different, estimated at 41% vs 29% per student in modern instruction.
A shock absorber reduces suspension rebound incidents. That's why they exist in the first place, to allow the suspension to stabilize rather than play a bouncy castle.
It doesn't eliminate other challenges in conventional landing gear, such as nose-downs. Engine thrust can't topple most tricycles, but it only takes one bad call with a tailwheel. You can see the runway straight ahead when landing a tricycle, with CLG it's just the sides at best, if it's a wide one.
In modern practice, almost no one flies a taildragger unless they're an enthusiast or just have to. The latter case is rough terrain or extremely short strips, which CLG handles better. Bush flying is inherently difficult, resulting in more incidents whatever the landing gear. However, overall serious injury rates are comparable, mostly due to CLG's lower landing speeds.
Shock absorbers are practical, available, and common on modern taildraggers. The only reason they're not universal is that some old planes didn't come with them. Planes flown in the bush today tend to have shocks.
Supplemental to Therac's answer, the Oleo strut was invented in 1915 and in use for aircraft by 1926. Suspension prior to that tended to rely on structural spring elements or elastic cords which had little dampening and certainly pilot memoirs and stories of the first world war and inter war era are littered with incidents and accidents where bouncing on touchdown was a feature/cause.
In particular attempting a three point landing at a high descent rate that ends in bouncing main wheels but not tail will tend to buck up into a high angle of attack/low speed situation where some combination of stall and/or pilot induced oscillation can cause a crash.
Taildraggers are dynamically unstable when rolling, because the fixed wheels are ahead of the center of mass. Think of pushing a shopping cart backwards; it wants to switch ends unless you actively control it from the end with the casters.
Factors that make taildraggers easier to control are more related to distances (arms) and angles that make the work of the steering element easier, than to the suspension system (although oleos are preferable to spring steel, mostly for a better ride and control of bounces).
Things that make the work of the tailwheel easier:
Wide track between main gears is better.
Ample rudder power is better.
Ample steering traction and authority of the tailwheel itself is better.
A long arm from the main wheels to the tail wheel is better.
Main wheels close to the center of mass, longitudinally, is better to reduce inertial moments, but this has to be balanced against nose-over tendency when braking and the allowable CofG range.
And a very important factor, not well understood: wheel alignment. Taildraggers must NOT have toe-in on the main wheels. Neutral to slightly toed out is essential. When a taildragger turns, weight is transferred by centrifugal force to create an outer wheel traction bias. The increase in traction on that wheel, if there is toe-in, tends to tighten the turn. If there is toe-out, it tends to widen the turn, reducing the instability. Alignment must achieve neutral or toe out when the gear is flexed by bumps and braking so the value used for rigging the main wheel axles (often done with shims) will take that into account.
Plenty of pilots with short coupled homebuilt taildraggers, like the Pitts Special, became afraid of them because of their extremely squirrely behaviour, when the real problem was just that they had mistakenly dialed toe-in to the main wheels when they aligned the main landing gear.
