Yes, but that's a meaningless slogan. "People" can't be in power in general, only individuals can, that is the nature of power. And so when it comes to actual implementation time, socialists put specific individuals in power - often a lot of power - and they spend a lot of time trying to ensure those people are good/moral/intelligent/wise/etc.
This is why left wing movements have a habit of becoming obsessed with specific leaders. Mao, Stalin, Chavez, and in more modern times Jeremy Corbyn, to a much lesser extent Bernie Sanders etc.
You later assert that "dictatorship of the proletariat" doesn't actually mean dictatorship in the e.g. Soviet sense even though that's exactly what it always turns into. Because, again, dictatorship of the people is a meaningless slogan.
[Marx] with Engels collected hundreds of factory reports and studied them for theirty years in nearly abject poverty
I'm well aware of Marx's life story, that's why I said it was cloistered.
He spent his entire life doing nothing but reading books in his library and writing political philosophy, sponging off his friends to make ends meet. For a man who claimed to think about the poor so much, here are some notable things he appears to have never done:
* Visited a factory
* Gone down a mine
* Socialised with the working class
* Attempted to win votes or take part in any kind of political process at all
Basically, his life was devoid of reality checks of any sort, which is why his ideas turned out to be so useless and disastrous. He even turned down an invitation from Engels to visit his factory, iirc - that's how much he cared about real workers!
This entirely sealed off existence is one reason he ended up engaging in so much intellectual fraud. His books were notorious even in his lifetime for false claims and faked citations. In particular he frequently found himself making false claims that industrial capitalism was in a downward spiral, because he was citing decades old reports from factory inspectors. Those reports had gone on to trigger changes in legislation and factory conditions, which later improved. His entire thesis was that industrial capitalism would have an apocalyptic end, but it simply wasn't happening, in fact the opposite was happening.
If he had actually spent any time doing real work, or had even joined the factory inspectorate himself, he could have observed this improvement directly. But he lived entirely through the experiences of others, in a dreamworld of his own making.
This seems to more describe a very limited range of people (as much of a range as a binary separation can convey) oriented entirely around the center of the political spectrum. The Socialist left dosen't talk about finding the right people to put in power, it talks about putting the people in power, over capital. They have no time to talk about taxes or tradeoffs or redistribution - to them, these are nothing but empty gestures that do nothing to change the power relations that make up the world.
I (and many in contemporary philosophy, sociology and anthropology) disagree with the assessment that Marxism is necessarily a "nonsensical" set of beliefs; we see it as a mode of analysis that must be constantly subjected to questioning (hence the origin of the Frankfurt School, post-Marxism, anarcho-Communism etc.). He didn't believe that all flaws stemmed from capitalism, rather, he believed in the materialist doctrine of history - that changed circumstances produce changed men (with some caveats).
Not only would Marx have markets abolished, he would render them useless; the market requires capital and money, it requires private property and all the relations that will drag with it - the double separation of the worker: separated from the means of production, and from the products of production. If this double separation is abolished, then markets can no longer exist.
But the dictatorship of the proletariat isn't a "dictatorship" as you may understand the word in terms of Hitler or Stalin, it literally means that society is under the dictation (direction) of the proletariat, not that there is some elite group which takes control and dictates things. The "wrapping up" refers to the concept of withering away of the state, which would happen in the case where the functions of the state have been transferred to the people. But there are other theories too, such as Kropotkin's anarcho-Communism.
But under no circumstance can you claim Marx lived a cloistered life; having the pressure of his father in particular weigh down on him to study Law for money, he chose philosophy, then exiled from his own country for publishing newspapers, he fled to London and with Engels collected hundreds of factory reports and studied them for theirty years in nearly abject poverty. This man was the very opposite of cloistered. But plenty of non-cloistered people have adopted his views and indeed many have been theorists too - check out Robert Tressel.