No, not least because Rome isn't remotely recognisable as a welfare state. It had corn laws to try to ensure the poorest were fed because otherwise they were inclined to riot.
It's about 150 years old. That could be viewed as old or new, depending on perspective.
Small hunter-gatherer societies were not socialism any more than ancient Rome was capitalism. Which is to say, there are certain parallels that can be drawn - similarities in certain forms (such as social arrangements and ownership systems) which can be used to illustrate concepts - but they're not really the same thing.
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How can socialism work? Not necessarily via government diktat. The aim is to increase pay. The government could do it centrally by setting salaries (e.g. minimum wage), but it can also empower labour unions and let the workers fight for themselves (why else do businesses and right-wingers spend their time trying to restrict unions?) It could do similarly by mandating worker representation on corporate boards, again giving workers the means to fight for themselves. More potently, it could mandate that a significant and minimum proportion of corporate shares must be owned by workers, thereby giving the workforce some ability to both profit from and control the company, seeing as they part-own it. It could encourage the creation of co-operatives, and so on. Thus there are many ways government can achieve socialistic policy by empowering people rather than itself. It is these sorts of things that modern socialists more look at. Bar the tiny remnants of Stalinist / Maoist cranks, basically no-one wants the USSR back any more.
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Places end up as autocracies or authoritarian basically because of instability and some sort of catastrophic imbalance of power, which may or may not be economic. The collapse of the Roman Republic is essentially that of social political instability where the major players increasingly understood that the newly-created professional army held the final say, and thus increasingly reached for to settle political disagreement. Imperial Rome was the logical end point: under the veneer of legitimacy, just a military dictatorship. It's not really anything to do with "capitalism" or "socialism" or "welfare" - it was just a grubby, winner-takes-all power squabble between different nobles. Capitalism can and will end in the same for the same sorts of reasons of imbalance and unchecked power-seeking.
To this end, one thing you can look at is something called "neofeudalism", which is a theorised point our modern capitalism may reach. The increasing concentration of wealth ownership in the rich, who extract rents from the poor; the capture of government by the elites and corporations, who then also make the law preferentially work for them, decreased social mobility, and so on. It's quite plausible.
