By bureaucracy, I mean that important state decisions are made by officials elected at best by their own party. Does Marx approve of this? Do any Marxists not approve of it?
It’s possible that someone or some Party that was over-influenced by Engels might harbor an interest in bureaucracy. See Rubel on Karl Marx: Five Essays by Maximilien Rubel (Editor) Hardcover, 288 pages Published July 31st 1981 by Cambridge University Press
Marx would as soon see the State and State apparatus disappear as soon as possible. Again, Rubel: Maximilien Rubel 1973, Marx, theoretician of anarchism https://www.marxists.org/archive/rubel/1973/marx-anarchism.htm
Why do I keep mentioning Maximilien Rubel? Because he was one of the few people (and scholars) who actually read probably all of the works of Karl Marx.
A lot of young people today, and even older people first learning Marx, are likely to see the Soviet system as something important. It was not. I see it only as an example of what NOT to do.
Going back before Stalin, Lenin did prove himself to be a very capable philosopher with his own personal study of Hegel and his notes thereon.
Marx and government
On the one hand, Marx saw government as a tool of oppression, by which one class dominates the other (classes.) In this sense, government should not exist in the ideal world (ideal according to Marx - i.e. under Communism). The power was supposed to originate directly from workers - but how exactly, he never explained.
Council communism and anarchism
More concrete proposals come from Communist anarchists (like Bakunin) and later Marxists like Gramsci. Anarchists envisioned a system where the communities of different level would decide on all issues in assemblies or via referendums. This approach has been criticized as based on an assumption that people can always reach a consensus on any issue, and as too unwieldy to run big industrial projects - e.g., railways. Indeed, Bakunin and Kropotkin were certainly inspired by their knowledge of the peasant communities in Russia at that time, which were hardly industrialized.
Gramsci, writing already after the Socialist revolution in Russia, envisioned a system of councils of different levels, where the representatives of higher level councils would be elected at the lower levels, with the power of recall at any time. The difference of such council system from an elected government is the diffuse nature of power - the higher levels of power never deal with the questions that can be solved on lower levels. This is not unlike the Federal system in the US or Germany, where the States/Lands have significant autonomy from the central authority - as opposed, e.g., to a more centralized government in France.
Revolutionary vanguard and Dictatorship of the proletariat
Although Marx never spelled out how he saw administrating life of the communist society, he clearly pointed out that the immediate transition to such a society was impossible. This gave rise to the notion of the dictatorship of the proletariat - the period where the proletariat, as the leading force of the communist revolution, would exercise dictatorial powers aimed at destroying the existing way of life, mode of production, culture, etc. - all to be replace by the new ones, corresponding to a classless society.
It was however clear that proletariat, even though not a representative of all classes, was still too large a group for making decisions. An important issue was the lack of qualifications (revolutionary consciousness) and basic training in Marxist principles among most of the proletariat - at least, not on the eve of the revolution. This would be resolved via the concept of the revolutionary vanguard - a smaller group of proletarians or intellectuals, whose role was to prepare the revolution, educate the proletariat, and manage affairs in the post-revolutionary period. This revolutionary vanguard is shorter referred to as the party, although Marx probably never used this latter term himself.
In this respect, Soviet system was much closer to the original Marx' thinking than any of the proposed alternatives. One can accuse the Bolsheviks in trying to implement Marxist concepts in a country that was not yet ready for the revolution, and in trying to rush through various stages of capitalism and socialism, preceding Communism. However, one cannot accuse the Bolsheviks in being unfaithful to Marx.
It is then reasonable to ask, whether Marxism is a viable theory of society: On the one hand, the Bolshevik party became a New class, oppressing the majority of Russians, and enjoying higher standard of life, although officially not possessing any wealth. Alternatively, the communist revolutions have never taken place (not yet at least) in the developed capitalist world, where the Socialist and Communist parties mostly followed the "revisionist" path - openly deviating from Marx in terms of engaging in parliamentary politics, seeking to develop arrangement with the capitalist classes and the State via welfare system, and eventually abandoning Marxism altogether in favor of liberal ideology.