In larger aircraft, shallow, high-power approaches are performed as a matter of routine, but in light aircraft it seems to be frowned upon, especially in clear weather and smooth fields. Why is "dragging it in" considered bad in small aircraft?
For the GA fleet, there is some historical precedent operating here. Years ago, engines were much less reliable than they are today. Dragging it in generally means that you cannot glide it in if you have a complete power failure. This translates to saying in the event of an engine failure, you are landing off airport with limited selection of landing sites (your altitude is low as you are now below pattern altitude). So if you are flying behind an old Curtis OX-5 in your Jenny, a gliding approach to landing was recommended back in the day.
Engines are much more reliable today. After an analysis of landing accidents, the FAA determined that a stabilized approach to landing is less likely to result in an accident, and they changed their guidance accordingly. Stabilized approaches are partial power-on landings. Some old timers and some old time CFIs reject the FAA's statistical analysis and still teach/fly glide-it-in landings. Properly executed, both approaches to landing are safe. It is just that the FAA thinks that stabilized approaches are safer given engines commonly in use today.
It is also true that virtually all airlines require stabilized approaches in their operating specs, and that a go around is mandatory if the approach is not stabilized. With this in mind, flight schools that aspire to send their graduates to the majors will all teach stabilized approaches.
This is less of a "big -vs- small" question, and more of a "single-engine -vs- multi-engine" question.
So, to answer your question with another question: What would happen if an engine were to fail during the approach in each case?
In a single-engine aircraft you want to always be in a position to land if the engine fails, so you don't want to get too far from the runway, while in a multi-engine aircraft you can continue the approach if an engine fails.
You want to be careful not to take this concept too far though.
Also, in jet aircraft the power needs to be kept relatively high so that additional power is available quickly if needed since the engines take awhile to accelerate.
Partially because of this, the idea of a stabilized approach was widely incorporated. This requires that the aircraft is in a stable, continuous rate descent to the runway and that the approach be abandoned if not within required parameters at a specified minimum altitude.
A secondary reason is that "larger" aircraft are commonly flying instrument approaches to help transition from the enroute/terminal phase of flight to the runway. These standard procedures are designed for use in instrument conditions and allow plenty of time to safely configure for a stabilized landing. Even in visual conditions, they are commonly used when there is a lot of traffic to "put everybody in line". Even a small single-engine airplane will probably be further from the runway than they like at big airports, just because of the amount of other traffic.
Actually a power on stabilized approach makes sense. Being close the runway in case power fails does not. Else you would never leave the pattern! After a cross country wherein my engine has been running just fine, why would it all of a sudden be an issue when I land? Now if you do have issues, carb ice, oil tempt, etc, I understand the safety margin. Also it is Good for “off field” engine out practice. So I do both, in the Citabria I glide it in from downwind abreast the numbers. Skyhawk SP I do a power assisted approach. Mostly because that is what you do in IFR flying, it’s good practice even when vfr. That is the main reason big guys power it in. Same procedure every time regardless of external conditions builds consistency and kiss! Also turbine engines are extremely reliable! Remember, no matter what you choose, a stabilized approach after a stable pattern is the foundation of good to great landings. I am not a cfi, so this is merely my experience not instruction!