Yes, there is a contradiction between them as you underline.
However, nothing about FDR's "economic rights" says how they should have been implemented.
Read them and say where it mentions redistribution.
The economic dimension is kind of the Achilles heel of political aka negative rights in that they can be disrespected anyway.
One could in theory act indirectly through this medium in such a way as to go against your political rights.
In fact, this is happening today with bail-outs and the more recent bail-ins in such a way as some people have committed suicide following the loss of their savings.
In the specific case of healthcare it is possible that you end up in a life-threatening condition in such a way as your entire savings will not suffice to get treated. So you end up dying which is kind of dumb.
Due to this inelastic demand, it would in theory possible to have healthcare run according to a co-op model in such a way that it's out of the state's hands.
Also this guy explains why healthcare costs are so high.
https://www.youtube.com/watch?v=qSjGouBmo0M
If you are thinking along the lines of rewriting the UDHR then you should join the libertarian party.
Also, if you are full on against socialism according to the original meaning of the word, you should steer clear of any form of insurance as it is pseudo-socialist or mutualist.
Finally, you may not like Wikipedia but their articles are sourced. So read the sources in case of doubt.