Here is some brief testimony about these mercs.
Should the US continue to allow them to dominate West Africa and be a thorn in the Ukraine conflict?
They are a criminal organization that takes over small countries. They recruit prisoners because they themselves are criminals guilty of all sorts of atrocities. Time to dismantle them?
The USA does not exercise jurisdiction over e.g. Ukraine, Syria, Mali, etc. As such events in other countries involving Wagner-Group is de-facto none of the USA's business.
Due to this the US government is trying to classify them as a terrorist group/organization, which would allow the US government to take certain actions under the umbrella of anti-terrorist operations.
Wagner-Group legally forms as a private enterprise. That most of it's officers and NCO's are Russian Army regulars is known, also the fact that they are directly under the command of the Russian MoD.
Certain Special operations are conducted under the leadership of the FSB and GUR. However as long as there is no proof of direct involvement via specific orders from the Russian MoD e.g. go and slaughter some civilians or don't take prisoners - the MoD can't be held accountable for such actions. (deniability).
WG can't be hired via foreign private contractors but only by foreign government institutions - as such WG members operate under e.g. Mali's MoD - and unless the UN would decide otherwise, it would again be none of the USA's or any other countries business as to what WG aka the Mali MoD does in it's own country.
Most of the WG members are actually not infantry grunts, but specialists in all kind of fields, starting from security personal and instructor (qualified in all kinds of weapon and weapon support systems) right down to radar and electronics experts to pilots. As such the actual operational envelop of WG members is very similar to that of former Green Berets operations in South-East-Asia and Central America and those of e.g. Air America. I would even dare to say that's were the general idea derives from - same goes for various similar organizations throughout the world.
A successful attempt by the USA to classify WG as a terrorist group - would have huge ramifications throughout the world including the USA, in regards to a prosperous multi billion $ business.
In regards to Ukraine the deployment envelop of WG members is vastly different - see convicts and other scum more or less press-ganged into WG as grunts, but with a significantly higher pay then their pendants in the regular RF. Again deniability and maybe far more important - the Russian general public wouldn't know or even care much about those losses.
The present discussions in Russia are basically to separate the actual WG and it's qualified members and needed experts - from the press-gang grunts.