MAK

I have never heard bicycles with pedals being referred to as push bikes (although I have been known to push my bike up a hill or two &#128546 As I've grown up, pushbikes are either the pictured bikes often favored by Amish riders or strider and other brand children's bikes favored by young children. I'm only 68 yo. so perhaps it's a term from before my time.

Reynolds 

I'd ride on them, no problem. Would wear a hi viz vest or jacket though.

rhm

So would I. Lights on, all the time, night and day; reflective vest; reflective "slow moving vehicle" triangle on the back, and I'd probably ride on the shoulder a lot of the time. I always wear a helmet mirror, and I'd keep an eye on that all the time.

rm -rf

If I have to ride on a busy main highway, I prefer ones with a middle turn lane. It makes it easy for cars to move over when passing.
I would need to use very puncture resistant tires if I rode on a shoulder.

Yes, an eyeglass mounted mirror makes riding with traffic much easier. With just a small head movement, I can sweep the view all the way across, to see everything back there. It's great for merging for left turns, or seeing when there's no cars overtaking me.

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Strava Heat Map
You might get some alternate routes from this map.

This is a map that's color coded by how often a road is used by Strava riders. It's from two years of strava GPS recordings -- 36 million rides in the USA in 2017!
You can bookmark/favorite the map view, since the URL changes as you pan and zoom.
As you zoom in, it recolors the roads so that locally popular country roads are now brighter white. Country roads get fewer riders, so they look dim on a zoomed out view.
(On the left side menu, click the Bicycle icon, and click the Labels button, these settings don't reload with the URL.)

I use this map to look for popular roads, and to avoid dim red roads that have popular roads nearby -- cyclists are avoiding those. Some popular roads might be too busy during rush hour, mostly ridden on weekends or evenings.

The map near your mentioned roads, Limmer, University...
https://www.strava.com/heatmap#13.55...55310/hot/ride

It looks like there are a few alternate east-west roads, but some areas don't have any through roads.

Colors: the bright white is most popular, dim red least popular. Roads with no recorded bike traffic show gray.

noglider 

@MAK, I learned the term "push bike" when I traveled in the British Isles, and I guess they use it also in Australia. I don't hear it in the US, and I'll bet they don't use it in Canada, either.