Evidently you didn’t get to the fifty oneth way.
I didn’t use Intersect Faces at all. Instead I used Follow Me. I started by copying the curve profile from one of your straight sections as well as the bottom “oval” from that group. In order to get a workable path for Follow Me I used Offset to create the smaller oval inside. Note the vertical guideline is aligned with the top end of your profile and intersects the side of the small oval.
Then I selected the path, got Follow Me (I use a keyboard shortcut) and clicked on the profile.
FWIW, if you use the larger oval as the path, the resulting extrusion needs a bunch of cleanup that isn’t required with the way I show above.