Your MD 308 amp is known for a balanced presentation and is probably not the source of the harshness and shrillness (unless it hasn't been broken in for long enough). Your Rega Jupiter is probably also not the problem. Per your question, switching to a tube amp could smooth the highs somewhat, but I suspect the better answer would involve a combination of different speakers and some room treatment.
Alternatively, a cheap and dirty fix would be to buy an equalizer and use it to reduce the treble energy. This is pretty sure to work, but you have to accept the compromise in signal quality that would be incurred.
Alternatively, a cheap and dirty fix would be to buy an equalizer and use it to reduce the treble energy. This is pretty sure to work, but you have to accept the compromise in signal quality that would be incurred.