No. He never wrote on ways government should work. Again, there have only been interpretations from others after his death.
His masterwork Capital damned the bourgeoisie class, a class he was a part of, his ideas threw him into poverty and yet he never stopped publishing his works, despite his situation. Seems like his "political agenda" was actually giving a fuck about the majority of the populace, the working classes, of all nations.
Marx premise was everything involved with the production of a product less wages was profit. Therefore the only way one could have a surplus/profit would be by decreasing wages.
Basically Marx believed capitalists did not give workers the value they actually produced, capitalists stole it in surplus value. This is pretty much unavoidable since if capitalists actually gave workers their fair share, the capitalist class would cease to exist as all their wealth is produced by the worker. This is true, it's why workers get paid wages and capitalists have inordinate amounts of wealth, despite not producing anything.
This is totally false as you can take the lowering of costs and maintaining a steady wage and still see lower profits.
This is just an attribute of a shitty business and not a rebuttal to anything Marx wrote.
His economic approach only takes into account systemic company risks and completely disregards market risk, or factors that said company can't control. A company can not control the weather, a government/group of controlling companies flooding the market with supplies destroying companies "profit" (example OPEC)
Marx lived through several of capitalism's inherent recessions and wrote about capitalists competing with other capitalists, so yea he knew these things.
Hell Karl Marx formed many of his ideas from Adam Smith
Yeah he read The Wealth of Nations and other bourgie economists. This is partly why he understood capitalism so well. He obliterated many of Smith's arguments and developed a better, more concise labour theory of value than Smith did.
hence unions
Unions haven't helped improved overall working conditions adequately if at all, unions sure as shit aren't helping working class in the third world. This is mainly because of the push back from capitalists, hence why capitalists and capitalism need to go if workers everywhere are ever going to be truly free.