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The benefit that a free market is expected to give to society is, as far as I know, that it will cause a "survival of the fittest" scenario in which the best solutions for a problem gain widespread acceptance and bad solutions die out.

Not at all the case. You're injecting Social Darwinist assumptions. The benefit of a free market is in having individuals coordinate their plans most efficiently based on their subjective preferences and valuations, with prices as an epistemic vehicle for conveying notions of scarcity, opportunity for arbitrage or other criteria. It is thus a continuous and dynamic disequilibrium process that one cannot anticipate through outside observation, as optimal outcomes are by definition always being revised, renewed and discovered endogenously.

But that of course only works if there is actual competition - which in many fields isn't the case in our current situation.

Competitiveness is one aspect, but a much more important one is constestability. Free markets are by definition ones allowing for the greatest degree of it. And I absolutely agree that our current situation is far from optimal. We are not discussing our current situation, however. Monopolies are a multifaceted subject, many of them based on rents obtained from IP law and hardly "natural".

Similar thinks can happen with large-scale accumulation of property.

There are things called diseconomies of scale that put upper bounds on the profitability and viability of being too big. Economic calculation issues set in, principal-agent inefficiencies, coordination problems and so forth. It is only through the subsidy of an exogenous body such as government that one can hobble along in inefficacy for exorbitantly long times.