I've used a variety of radios and amps over the years including tube/analog, SS/analog, analog/digtal hybrid and SDR, including mixing tube and SS amp/transceivers. I've never damaged an amplifier by initial key down overshoot, though I've seen TX power overshoot to one degree or another in just about every case.
And I have usually used ALC, but adjusted it by various means (sometime HW changes) to avoid kicking in unless the amp was well into overdrive.
Earlier SS amps were more prone to damage from many causes, so perhaps some were vulnerable to this. My Yaesu Quadra was not, but did have a pretty poor, slow-acting (CPU driven) output protection circuit. My SB-200 tube amp didn't care, either - it was most vulnerable to duty cycle and was really best used for SSB.
Some ceramic tube amps were also suseptible to overdrive damage - but this usually had to be sustained, not a brief blip.
Currently I've used my own or friends Yaesu FTDX-1200, -3000, -991 and Icom IC-7300 with KPA-500 amps. They ALL seem to have initial key-down overshoot, per my attached LP-700 wattmeter's TX 'scope trace. The KPA-500 does not seem to care at all, either. But I rarely push it beyond its rated 500W.
In my experience, momentary overdrive peaks on TX are not usually a problem for a well designed HF amplifier, as previously pointed out. I suppose its always possible to make it a problem by using no ALC at all and overdriving an older SS amp or ceramic tube amp without fast input/output protection. But there is always a way to damage equipment if one is careless or deliberately pushing it too hard. No stopping that.
Brian - K6BRN