Republicans are anti-Russian from a
military perspective, because some of their traditional interests are rooted in that industrial area and will always benefit from a heightened state of military urgency. That's the "John Bolton" wing, basically, which is mad at Trump for squandering their best opportunities of payday (you can't sell weaponry without an enemy, and it's hard to get a credible enemy when the key guy is so friendly to all the best candidates). That drives a lot of their other objections to Putin's reach.
From an economic perspective, imho they display the same oligopolistic tendencies that Putin's Russia exhibits: lip service to competition and open markets (mostly abroad), ruthless devotion to monopolistic positions and cartels at home. The definition of "free markets" is declined differently depending on circumstances, in opportunistic fashion. I think that's what you mean by "competitive power structures" - they are competing entities, sure, but I personally don't see them as particularly different in the way they behave. Trump lays this bare by dropping the hypocrisy: him and Putin speak the same language, in the same way Putin and Berlusconi did. Both Trump and Berlusconi spent their lives in the trenches of "the Market". I don't think this is a coincidence.
I also disagree on the view of Europe being significantly more anti-corporatist than the US, but I admit that my Anglo-Italian view might be excessively pessimistic in that area and it's a complex discussion anyway, so I won't go there. However, the difference in "softness" towards Putin imho is not there - the Eastern block in the EU, for example, is rabidly anti-Russian for understandable reasons. We simply know that we cannot live without Russian energy at the moment, and that results in some hard pragmatism.