if part of that feature is to more slowly ramp up the application of power, will that not also affect these diodes’ reaction speed overall and negate the reason for their substitution in the first place?Yes, to a certain degree. You don't have to worry about the tube rectifier going bad... For my part, I'd put the amp on the bench and see how bad things are when a set of HEXFREDs are installed. Measure the power supply voltages at idle, compare to the ratings of the caps (usually if you exceed 85% you're risking over-voltage under certain conditions).
One thing that can be problematic is the power supply is charged up long before the tubes are so there is no load. The no-load voltage is much higher! Some manufacturers (and I've no idea about Cary in this regard, I'm not calling them out), because a tube rectifier is used, will have lower voltage caps in the power supply as a result. But with a solid state rectifier, the caps will be in a no-load situation and so will be much higher than when the power tubes are warmed up.
If that no-load voltage is too high you can destroy the filter caps- and it won't just be the input caps to the power supply that are that high- **all** of the filter caps will be at this voltage until the tubes warm up. So you really want to be careful! It can be really upsetting when a filter cap blows its guts out.
If any of this sounds intimidating, good- if that's the case, you really might want to resist that urge to see what can be done. OTOH, if you feel like *perhaps* also replacing the filter caps maybe it won't be any problem. But at some point you might want to consider building the amp up from scratch.