What is a province?
We should start with the development of the provincial system. Once upon a time, the Romans came to a place (with allegedly defensive intentions\s Cic Rep 3.23.35) and started military operations there. Those operations had a commander: that commander, usually a consul – or person acting in place of one – had a task. That task was usually "go over there and defeat so-and-so". Or, in the case of Publius Cornelius Scipio in 205, command the army in Sicily with permission to go into Africa if it would be in the interest of the republic. Drogula Commanders and command (2015) p 285, citing Livy 28.45.8 (my loose translation).
This task is a provincia.
To be clear "defeat Macedon" is a provincia. So too is "defend this place against barbarians". And when Rome starts to assert direct rule over definable places (Sicily, Sardinia, and Macedon are the first), that task becomes "keep this place peaceful and not a pain in our assholes/s". Only by the start of the common era does provincia turn into something akin to modern English "province" referring to a place that is regularly administered. See Ricardson Language of empire (2008) p 8; Drogula "The concept of provincia in early Rome" in Commanders and command (2015) pp 131ff.
For much of the republican period, there don't really exist provinces in the way moderns think of them. These "provinces" are geographically separated – mostly to make it clear who, by default, has supreme command (imperium) where – and are fixed by anything other than convention. Drogula Commanders and command (2015) p 146. Their nature as fundamentally military commands also gives the Roman magistrate a huge latitude as to what he can do there...
...such as enslave or kill entire populations. See generally Baker Spare no one (2021). The senate is far away and the magistrate on the scene has supreme power; very little could be done to restrain him. Drogula supra p 137; Baker supra pp 185–88 (a magistrate in Spain during the Lusitanian wars had, after they surrendered, three tribes' entire male populations put to the sword with the women and children enslaved; on his return some men, including Cato the Elder, attempted a prosecution but it was dropped; the magistrate was elected consul about five years later).
How did provincia turn into "province"?
As more provinces became "administrative" in character ("go there and keep the peace"), military exigency became less an excuse and in many cases a peace settlement bound the magistrate by law. Roman governors, however, with their legions were still seen as offering authoritative judicial services with real enforcement. So what became more common then was a magistrate (usually indebted because he borrowed a ton of money winning his office) showing up in a province and demanding all sorts of gifts. All the better if sent to the east, where the cities are rich and "gifts" plentiful.
The Romans of the period did see this kind of behaviour as a bit too extreme. Extorting subjects is all well and good but if you squeeze them too hard they will revolt and crushing that revolt is, among other things, expensive. So, starting in 149 they created a permanently sitting court (quaestio perpetua) to try these extortion (repetundae) cases. Initial attempts, where the senate deal with them ad hoc, clearly were not sufficient. But requiring someone to come to Rome and sue for mere recovery of damages already taken as in the law of 149 was not really sufficient either. Drogula supra pp 276–81; cf Livy 26.32.
But strong legal restraints on what commanders can do only emerge in the late republic. These take the form of a somewhat bewildering net of legislation such as the lex Porcia (100 BC), lex Cornelia de repetundis, the lex Julia de pecuniis repetundis, the lex Julia de provinciis. In toto these established all sorts of prohibitions on what kinds and quantities of goods a magistrate could requisition, stay in defined geographic areas instead of starting wars without permission, record-keeping requirements to ward against extortion, along with enforcement at Rome in real courts with teeth. Drogula supra pp 285–87.
The passage of these kinds of restrictions both in Roman law and in treaties or postwar settlements, along with the clear links between the Roman aristocracy and provincial elites, also show that the Roman aristocrats did not see the provinces solely as places to extort money. Obviously, many of them did; so too did the equestrian publicani who saw the huge profits to be made from Asian tax farming. But they were also places to build influence and support (both among provincials and Romans living there). Especially in the east, people now living under Roman rule built relationships with Rome to pursue their own interests. See generally Gruen Hellenistic world (1984) (notably also concluding that the Romans had no overarching strategy of imperial expansion but acted largely ad hoc).
Now, after that long background:
By imperial strategies I mean for example seeing the provinces as a breadbasket for Italy, a market for Italian goods, a source of a certain class of soldiers, the personal playground of the highest senators, a far-off place to send dangerous senators, or something else entirely?
The Roman aristocracy saw the provinces mainly as places to win glory, money, and influence. Those things traded off against each other. And there were political, and eventually legal, constraints on how much glory, money, and influence someone could wring from the provinces.
Beyond that theme, all the things you list could be said of the provinces to some degree.
- Sicily was long seen as a place from which grain could be procured for Italy.
- Italian goods were regularly shipped abroad, but it is important also to recognise that the Romans were not very economically interventionist in a modern sense; they would have contented themselves from securing exceptions for Romans from port fees or the like.
- Certain kinds of auxiliaries could only be recruited from certain provinces.
- Some magistrates certainly played the role of tyrant when abroad.
- Senators convicted of crimes could be sent into exile in the provinces. Eg Publius Rutilius Rufus (falsely convicted of corruption for being too hard on the Romans in his province, he spent his exile living as an honoured guest of the city which he was convicted of extorting).
But, in the main, these are separate from the Roman politician's choice to go to the provinces. (One could decline: eg Cicero after his consulship in 63.) Senators usually accepted provincial commands to bolster their own careers. They saw them as places to, if you will, show their quality.
Tiberius Gracchus ... argued that the land reforms in favor of the peasants would produce a stronger peasant soldiery ... Tiberius proposed full citizenship for Italian, after all, not any provincials.
TSG, with the support of many senators, was acting to forestall the then-apparent indication that the population of Italy was falling. Putting more people on the land could have helped stop that. Roselaar Public land (2010) pp 229–30. Modern archaeology has shown that the population was not in fact falling. Potter "The Roman army and navy" in Cambridge companion to the Roman republic (2nd edn 2014) p 68.
The evidence that TSG wanted to give the Italians – contra Romans living in the ager Romanus not close to Rome – citizenship and land is weak. It's in Vell 2.2 but most scholars believe that Velleius is likely confusing Tiberius with Gaius, who did propose giving the allies citizenship. Tiberius' policies would have benefited Italians basically not at all since they would have been dispossessed of lands they currently used in favour of new Roman settlers. Roselaar supra pp 244f.
The relationship between the Romans and the Italians is another question (and book). I find Mouritsen Italian unification (1998) convincing. However, Mouritsen's position – that the Appianic narrative is wrong and that the Italians fought the Social war to overthrow Rome rather than for citizenship – is not without controversy.