What is the best Android GPS navigation app that lets you change the given route?

I don't use GPS.  Once when I did, it was going to take me 100 miles north of where I wanted to go.

Just to state up front, I agree that navigation applications in general aren't the most user friendly, nor do they always use the best route.  Various applications have their strengths and weaknesses.  It's been like this for years.

The way most of them work is they assume that there will be no traffic issues and that you will always be going the posted speed limit.  Using that as a basic premise, they determine the shortest route from a time perspective.

Using google maps you can add stops to a route, which can be leveraged to force a specific route.  First create the two end points on your route.  On the screen that shows starting and end points with an overview of the route map, tap the three vertical dots on the right.  There you can modify route options to say avoid tolls, you can also "add stops".  You could add a stop such as Verrazzzano-Narrows Bridge, which will force the route.  

Using your google account, you can login to google on a PC, than create the exact route you want (you can drag the route to whatever road you want), then text it to your phone.   The problem is you need to plan all this in advance, regardless if you use the phone to setup the route or a PC.  It's nearly impossible to drive and set the route simultaneously.   

In my opinion, google maps is the current gold standard for mapping applications, at least for the USA.