Why would that article list a disadvantage of regulated power supplies as "The amplifier can no longer deliver higher power on transient peaks."?
The reason is as follows: as I stated, an unregulated power supply's voltage will drop to some degree as the amp draws more power from it. So, for example, suppose we have an amplifier with unregulated power supply which is rated at 100 watts into 8 ohms, and another amplifier with regulated power supply which is rated the same.
For the unregulated amp, the power supply is +50/-50 volts, which decreases to +/-42 volts at full power. For the regulated amp, the power supply is +42/-42 volts all the time.
Now, since music usually does not play at full power, normally the unregulated amp's power supply sits at close to its quiescent level, which is +50/-50 volts. Along comes a transient. Initially, the unregulated amp is able to put out more than 100 watts - say 130 watts - because it has higher power supply voltages - however, as time goes on (we are talking 10s of milliseconds here), the power supply voltage sags down to +42/-42 volts and the amplifier can only put out 100 watts after a short interval. Thus, the unregulated amplifier is able to put out a higher power for a short period of time, but eventually physics catches up with it and it can only supply 100 watts on a continuous basis. Once the transient has passed, the power supply can build itself up to +50/-50 volts again for the next transient. Note that this change in power supply voltage with the transient will affect, to some degree, the ability of the unregulated amp's ability to faithfully amplify the signal. An obvious way this happens is if the transient continues at the same level, it's initial peak will be reproduced at 130 watts but its continuation will only be reproduced at 100 watts as the amp will clip during the latter part of the continuation.
On the other hand, the amp with regulated power supply, because it starts with lower power supply voltages, can only put out 100 watts - but, it can put it out on a short term basis, or continuously. That is why an unregulated power supply allows an amp to put out more power than its continuous rating. So the regulated amp will clip on the first part of a transient at a slightly lower level than the unregulated amp, but as long as you play it below clipping, it will reproduce a prolonged transient more faithfully.
A classic example of this behavior is the original NAD 3020, which was rated at 20 watts/channel continuous into 8 ohms, but could put out 40 watts into transients for a short period of time. However, if you were to take the exact same circuit but run it with a regulated power supply set at the same quiescent voltages as the original design, it could have been rated at 40 watts continuous power. It would also have cost much more for the regulated power supply, and also probably for larger heatsinks to tolerate it being run at 40 watts. The original 3020 is an example of cost-effective design using a "soft" unregulated power supply, and takes advantage of two facts: first, that transients are brief and relatively rare occurrences and second, that the ear is relatively tolerant of distortion on peaks. The significant words here are "cost-effective." For example, the Mark Levinson ML-2 was also rated at 25 watts/channel, however, it had a regulated power supply, ran in class A, weighed 65 lbs for each mono amp, could put 100 watts/channel into 2 ohms, and cost a lot more. Consensus opinion is that the NAD sounds very good for the price. The ML2 sounds better.