The SHC is a nice entry level road bike from days gone bye. Though nicely finished, Sekines of the SHC's quality level offer little more than a nice recreational ride quality, coupled with fairly nice and very functional Old School component group. However...

Since you have come to appreciate the different ride quality, compared to the three speed (roadster?) mentioned, of the Sekine, then think about the next level of ride quality. A ride quality offered by a higher end vintage road bicycle will, not sure how to best put this, blow your kilt up when you take a quality road bike, that fits you properly, out for a spin. So...

The cost to upgrade the Sekine, in my opinion, would be much better spent buying another old road bike. With luck, and perhaps a better understanding of How To Find Vintage Bicycles, you can come up with something that not only blows the kilt up, but knocks your socks off at the same time. Or...

If you seek workout opportunity, perhaps a Sekine Single Speed, the conversion of which would be a cost effective way to keep the bike and have something a bit different to ride, now and again...