In my opinion, for a bike used on a trainer only, upgrading parts is not worth the expense. It's not like you risk getting dropped if you botch a shift. If you can't fight the urge to upgrade, buy something new for your on-road bike and transfer the "old" stuff to the trainer bike if you feel like it.

The quick shifting of newer bikes is mostly due to improved designs on the chainrings and cassettes, not the deraillers or shifters. The latter help but not nearly as much as the former, at least in my experience.

Of course, proper tuning is a must for any derailler system. Find someone who knows how to tune a bike and get them to check over what you've done. You might also benefit from new cables and housing.

Oh, and downtube shifters are not picky. Set to friction mode, they'll shift just about anything, including the set ups you mentioned above.