Is it possible that derated thrust takeoffs are safer than normal takeoffs?
The reasons presented at ground school for the benefits of de-rated thrust takeoffs were twofold:


less chance of catastrophic engine failure during takeoff
less overall wear between overhauls 


It is from this standpoint that reduced thrust takeoffs are safer.

As you point out, these are only authorized when the weights, temperature and runway length allow, but for the EMB-145 virtually every takeoff we did was eligible for reduced thrust.

You point about ease of control with an engine out is countered by engine out procedures which involve setting max thrust on the remaining engine (for two-engine ops).  During takeoff roll in the EMB-145XR, this puts the other engine at 117% power.  With the exception of flying near $V_{MC}$ (for light twins) the extra thrust is a benefit and not a hindrance to control and if this is at or after $V_1$ you'll want the thrust to go flying.

Your question about a bird strike is going to go one of two ways:


the bird hits the N1 fan and goes through the bypass ducting
the bird hits the N1 fan and is ingested into the N2 compressor stages


In either scenario a reduced thrust vs full thrust takeoff is not going to make a huge difference, but scenario 2 will be far more destructive to the engine than scenario 1.