I don't think Bismarckian statism is well suited to America to be honest. I mean the umbrella concept of a "nationalist economics" (at least to the extent of national responsibility, not necessarily outright protectionism) and coupled with the "desire to ease the strain on the system for the sake of its own preservation" are necessary yes, but the exact means of how you achieve that can be very different.
For instance against a backdrop of economic nationalism you have a choice between embracing monopolies, which than then be state controlled/influenced and move down the road towards state capitalism with the government redistributing through statism to keep the system afloat and keep the pitch forks at bay. That is very much the German economic model from the late 19th and early 20th century and we certainly know what that helped to lead to.
One the other hand, you can go the opposite direction while still under the umbrella of the "nationalist economic model" and the "the desire to ease the strain on the system for self preservation", by breaking up monopolies and trying to restore market competition. This was more the approach of TR, at least in the 1900's.
Ironically comparing TR and the Germans, I would argue the Germans came up with a more market friendly national healthcare system while TR's proposals were less so but of course TR never succeeded in implementing anything. I still think the German model for health care is the best starting point for the US, to which the additions of greater choice and some level of state influence/administration would make it well suited to our system. Certainly a far better match for America and its political history then going for a system like that which dominates the anglo-sphere.
The great trick in getting Republican support for healthcare won’t be support for a European style healthcare system. Instead, if you really want to get Republican support, draw parallels between a universal healthcare system and Israel’s healthcare system. That alone would break down a lot of opposition from evangelical and neoconservative elements in the party, which, combined, have enough of a base to be a solid majority in the GOP.