I have a Schwinn Racer that closely resembles yours. When I got the bike it was coaster brake. I drilled it to fit a front handbrake, and that was a nice supplement.

Over time, I decided a three-speed conversion would be nice, as I had a 26"x1-3/8" rear wheel from a Japanese-made common 3-speed bike, with a good Shimano hub. I also had a proper shifter and cable for the installation. Everything fit perfectly. It works very well indeed, a real step up from one speed for a casual rider like me.

Unfortunately, the newer hub wasn't one of the 3-speed-plus-coaster brake type, so I now had no rear brake. The junkpile yielded a handbrake for the rear, which was easily installed after I drilled the bridge.

Now the bike was much nicer to ride, but I'd given up much of the Schwinn's originality. As a final straw, I replaced the front wheel with another 26"x1-3/8" standard type. So now, the only Schwinn mechanical parts remaining are the headset and bottom bracket. Its appearance isn't too different from a 3-speed Racer model, though.

I don't recommend you do as I've done. A five mile commute doesn't require any changes in the bike, except for those you desire to make. You could just find tires that fit and enjoy the ride.
Best of luck in whatever you decide on.