All of the ships you linked to are operated by Rosatom or its predecessors.

That is essentially a state operator as far as I can tell.

State != military, though. For instance, all France nuclear power plants are operated by Électricité de France (EDF) that is substantially owned by the French Government, with around 85% shares in government hands ([1]).

I don't see a problem having nuclear container ships to be operated by civilian governmental agencies / state-owned enterprises. This might not work for the US, but we often have very unique ways of doing things.

1. https://en.wikipedia.org/wiki/Nuclear_power_in_France

i suppose these ships only use Soviet / Russian ports, and mostly don't operate in high seas. Icebreakers do operate in high seas, but getting close to them is a bit hard due to ice and polar weather.

The more interesting example is NS Savannah [1], which indeed crossed oceans unattended, and was accepted at European ports.

[1]: https://en.m.wikipedia.org/wiki/NS_Savannah

Of those, the icebreakers have been commercially successful due to logistics of fueling in the attic and the heavy fuel needs of ice breaking (both solved by a reactor that only needs to be refueled once every few years).