From my perspective as a pilot, we've used reduced thrust where able in most types of aircraft I've flown, certainly most all turbine equipment, including agricultural, corporate/charter airplanes, and airline equipment. It's been a practice in most types of operations I've flown, from firefighting to ambulance to crop dusting/ag, to cargo, charter, corporate, government, and yes, airline. Where it's safe to do, allows ample takeoff and stopping margin, allows adequate obstacle clearance etc, it's perfectly acceptable.
A reduced thrust takeoff means that one always has the option of pushing up the power as required, though all the performance calculations take into account climb gradients, going, stopping, and obstacle clearance without having to do so. This includes an engine failure; when we calculate reduced thrust takeoff performance, the performance data assumes losing an engine and continuing the takeoff...still at reduced power, still able to make the required gradients. Stopping is a no-brainer; the power will be retarded, ground spoilers deployed, and the aircraft stopped on RTO brakes where installed, or manual braking. Not rocket science, and it's all factored in...without reverse I might add. Reverse only shortens that distance.
Our operations manual spells out exactly when a reduced thrust takeoff can be used, and when it can't. Every reduced thrust takeoff is planned with the specific runway and runway conditions in mind, including any appliable NOTAMS such as temporary obstacles or reductions in length. Every takeoff is planned with an engine failure in mind, as is the departure path after takeoff. Nothing is left to chance.
We utilize reduced thrust takeoffs, and reduced power climbs as part of the nearly universal standard noise abatement departure procedures. We also have reduced climb thrust above 10,000'.
From a mechanic's perspective, reduced engine temperatures make for substantial increases in engine and component life. I've been an aircraft mechanic and inspector as long as I've been a pilot, going back to my early teenage years. I've been working on large radial engines, small pistons, turboprops, turbojets and turbofans for several decades now. I've had these engines apart, boroscoped them, handled every internal part as a regular function of inspection and repair. The differences in operating techniques or procedures show up in burned blades, cracked cans, metal creep, etc. A ten percent reduction in power equates roughly to a ten percent increase in engine life. If this can be done safely, all the more power to the operator...increased engine life also equates to improved engine safety, longevity of components, increased mean time between failures for stressed and hard use items such as turbine wheels and blades, etc.
A common method of operating reduced thrust is to use an assumed temperature. This isn't a wild idea made up by flight crews, but comes directly from the engine and airframe manufacturers after ample testing and design. One assumes a takeoff at a much higher density altitude based on a plethora of criteria and data, and determines if the aircraft could be safely flown off the current runway under those conditions. If it could still do so, still meeting all go and stop criteria applicable to each segment of the takeoff, then it can also be taken off at a reduced power which replicates a takeoff at the higher assumed temperature. The performance data is recalculated using the reduced power to ensure it matches, and when all data adds up, the reduced power is established for the takeoff. Nothing precludes pushing up the power at any point in the takeoff where required, nor performing a full power takeoff if required. However, it isn't required, and where a reduced power takeoff is performed based on an assumed temperature calculation, an engine may be lost and the takeoff continued at that reduced power, and still meet all the takeoff criteria.