There is a consensus that the key problem of Italy is low productivity, due to several causes.
The following is a paper by Banca d’Italia (Italian central bank) about the problem of productivity in Italy:
Illegality and corruption are also an obstacle to growth.
The problem of their impact on productivity and growth is dealt with in the paper by Banca d’Italia in section 5.5 Business environment, p. 58
Bureaucracy, enforcement of contracts (Italian justice is very low) are negative factors. Moreover,
“the presence of criminal organizations, tax evasion and corruption
jeopardize the ability of markets to allocate resources efficiently,
and it does so through two main channels.
empirical evidence demonstrates that the consequences of corruption,
crime, and tax evasion on Italy’s productivity and growth are
sizable.” (p. 61)
The other problem is the big Government debt.
The following is a recent paper by some of the leading economists in Italy about the sustainability of Italian public debt and its impact on growth:
Il debito pubblico in Italia: perché è un problema e come se ne esce(2019)
(unfortunately I didn’t find an English translation).
From the abstract:
This paper discusses the macroeconomic effects of Government debt,
reviewing empirical evidence and the main mechanism through which a
high debt can cause a reduction of growth. High levels of debt, in
addition to representing a potential constraint on investments,
weakens the effectiveness of economic policy and the ability to
enforce counter-cyclical measures to stimulate demand.
As illustrated by the papers above (by some of the most authoritative research centers and economists in Italy), corruption and organized crime is only one factor among many other that are an obstacle to economic activity (mainly because it discourages investment, together with other environmental factors as bureaucracy).
The main problems, as illustrated above, are the various reasons of low productivity and the high public debt.
This is the real picture of the Italian economic system, and the picture in the question that describes a country suffocated by corruption and organized crime is not realistic.
Anyway, to say 'why Italy is (at least relatively) rich' one has to retrace all the economic history of Italy, at least since the Second World War.
The most important period, the 'root' of richness of Italy, has been the so-called economic miracle of the years since the end of fifties to the end of sixties:
Just some other few words to give a balanced representation of Italy economic situation, and point out that, as Italy isn’t a country suffocated by crime and corruption,
it is not a poor and backward economy living on tourism (as some comment suggested).
On the contrary, industrialization is an important reason of relative Italy’s richness.
The economic miracle was a period of industrialization:
In the north, beside the north-west ‘industrial triangle’ (Genova,
Torino, Milano), characterized by steel and engineering industries,
also the north-east triangle (Padova, Vicenza,
Treviso) becomes established, characterized by manufacturing activity, spread also in
Lombardia, Emilia Romagna, Toscana e Marche.
In this period of the history of Italy, until the late 1980s and
early 1990s, among the big industrial realities that drove the
economic boom we remember the automaker FIAT, the Montedison group,
the manufacturer of office machines Olivetti, the engineering group
Ansaldo and the steel group Ilva. (from Italian Wikipedia -Miracolo
economico italiano, my translation)
The economic history of Italy since then has led Italy to become the second industrial country in Europe, after Germany, at least until the last few years. Now it is possible that Italy has been surpassed by France, but it is controversial.
In 2019, the industrial European production, came for 30% from Germany, for 17% from Italy, for 12% from France and for 9% from Spain (Eurostat data), as illustrated in the graph below:$^1$
$^1$ And often we are talking about sectors of high technological content, as in pharmaceutical industry and in automotive industry, in particular car components. For example, the most important German car manufacturer, as BMW, Porsche, Audi, Volkswagen import many parts from Italy.
Electric motors of Taycan Porsche are produced by the Italian Marelli, the brakes of BMW Series 1 and 3 are by Brembo, headlights of Audi e-tron are by Marelli, wheel hubs of Porsche Cayenne are by Mazzucconi, steering dumpers of Volkswagen Tuareg are by the Turin’s company Cultaro and many others (source Automotive News and HIS Markit).
See the graph and the data in this article, about imports by Germany from Italy of parts for assembly, brakes, motors, gearboxes and so on.