Hi, Richard and Mary,
I have never had a holiday in Scotland that was less than 27 days when travelling with Mrs. A. The few times that I've gone solo, it has been for 16 days or more. Having said that. however, I have always rented a car. Like yourselves, I am a senior citizen.
I would say that using public transportation, you should have no trouble at all getting around in Scotland, although service can be sparse in some of the Western Highlands and Islands. There are some islands (Arran, Islay, and Harris/Lewis) where the bus service is excellent; others (Skye, for instance) where the bus service is not quite as good. However, most of Scotland is well served by coach, rail, plane, and ferry. If you have an itinerary in mind, one of our regular contributors, Stuart (isn31c), is an expert on all things regarding public transportation, and can provide you with a plethora of schedules.
If you can extend your holiday past 19 days, all the better.
Don't let driving on the left side of the road put you off hiring a car. About 85% of the cars for rent in Scotland are standard transmission. However, having an automatic transmission, which may cost a bit more, would mean that you could concentrate on the driving, not shifting gears.
In addition to public transportation, there are several tour companies which provide tours lasting from one day to two weeks. If you don't mind sharing your travels with strangers (which you'd do on public transportation anyway), that would be a good way to complement the bus/ferry/train part of your holiday.
Whatever you decide to do, you will love Scotland!
Very best wishes,