I'm a Moscovite, so this answer is bound to be biased.

Safety is a very relative notion, I perceive Moscow safer than quite a few cities I've been to in Western Europe and the US (or at least some of their neighborhoods). Furthermore, I've once been detained by the US police for several hours out of the blue, so... yes, unexpected things happen everywhere.

>  “I’ll get a rental car, take a taxi or hop on a train if necessary.”

That concern was somewhat reasonable, as you could come across a dishonest taxi driver who would want to exploit you asking for a much greater fare than necessary. I knew a person from the US who paid almost $100 (20 times more than the norm!) for a trip from the airport. Generally, taxis are quite a mess and if you do want to get a taxi, you should first find a reliable taxi company and order a car to your location. Uber and Yandex Taxi might be a good option as well, though I've never used them in Moscow.

The express trains are a great option if you don't have a car, they're extremely safe, reliable and quick. You can read up more about then on https://aeroexpress.ru/en/.

The cheapest way to get to the city would be buses and regular suburban trains (look up the direction on the airport website). In all my life I've never had issues with either of those, but they may be slow (depending on the traffic conditions) and not as comfy (not fun to ride during rush hours!).

All in all, all of the public transport (trains, metro, buses, trolleybuses and streetcars) in Moscow is very safe and cheap, even if not always fast and convenient.

> Has the security level for Americans truly degraded to the point that I shouldn’t be taking in the beauty of Moscow on my own, as I try to do in every city I visit, whether traveling on vacation or on business?

I consider it mostly nonsense. That said, you should adhere to the basic tourist wisdom, which is not to let others see you as a confused and helpless tourist who could easily be taken advantage of. If you're Caucasian and don't wear a red, white and blue baseball cap and a t-shirt with the Liberty statue, few people would suspect you to be a tourist from the US.

In conclusion, the succinct advice I would give to anyone visiting Moscow would be to not be afraid and visit whatever place you wanted to visit, as long as it's not a military base or something. If you fear the wolves, you'll never get to see the forest, as the old saying goes.
Use the public transport to get to places, it's often jam-packed, but cheap and fairly reliable. Visit the museums and theatres, try some of the local food, go for a hike in one of the nearest forests and national parks, or get out of the hustle and bustle of the city and see some of the suburbs.

Just be aware of your surroundings as everywhere else, don't flaunt expensive electronics in public, learn the Cyrillic alphabet because not all signs are translated, don't expect most people to speak English or be willing to go the distance to help strangers who don't "even" speak their language, and keep a phone of the embassy and your hosts at hand for the unlikely cases of emergency.