How would I process and output every possible route of a stream network?

Not a real answer, but more than a comment..

TauDEM should be able to do what you want, and QGIS provides plugin support for it. There is an existing question here on GIS.SE that you might want to review.

[Update/Response to comment] Ultimately your workflow will need to utilize several of the methods to go from a raw DEM to a dataset from which you can extract the watercourses meeting your Strahler order criteria. For example, filling pits, calculating flow direction, flow accumulation, etc. The GridNetwork method produces a Strahler order raster as one of the outputs and needs only the D8 flow direction raster as an input, and optionally, a shapefile of outlet candidates if you wanted to target your reaches that way, which I would if I had that large of a study area.

With respect to processing times, since your study area is basically a continent (lol) I'm guessing you're using a pretty coarse DEM, like 90m SRTM? Something nice about TauDEM is that it can handle parallelized execution using mpiexec. (A note here, I'd never heard of mpiexec prior to using TauDEM, but on my Ubuntu Linux system, it worked right out of the box without having to install anything.)

Obviously processing time is going to be very sensitive to differences between system resources, study area size, DEM resolution, etc. While I can't predict how long your processing times would be with TauDEM on an area that size, I can describe my most similar experience: I've been tinkering with a statewide (South Carolina) study area of 30m SRTM data. I forgot how long it needed to fill the pits, but processing the D8 flow direction raster on my laptop (16GB RAM) needed about 30 hours. After that processing times on subsequent tasks were pretty trivial, often just a few minutes. My processing region is also large (perhaps also laughably large), and I can further tell you TauDEM was the 3rd open source software I tested (I also tested WhiteboxTools and RichDEM), but TauDEM has demonstrated the best memory management for such a large job, which I think is 100% relevant to your project goal. Also, being able to set the number of parallel processing chores was nice because it definitely softens the processing time, but it also gave me the power to leave a few of my processors free ..you know, for the Zoom meetings.

In my case, getting started with TauDEM on an Ubuntu system proved to be pretty easy. I already had GDAL and its development libraries installed, and I built the TauDEM code from source (note the README.md file in the src folder). Plus I was able to run parallelized jobs out of the box using mpiexec. Finally, I'm using the Command Line utilities, rather than either the QGIS or the ArcGIS GUI implementations, so that saved me a step. If you're using Windows your experience getting started may have different speed bumps, however, it seems like TauDEM is actually packaged more for the Windows user, so perhaps that won't present any trouble.