Typically early Republican Rome raised armies by holding an annual rally of military "classes" based on wealth. Show up with you weapons, have a practise and maybe vote about going to war that year. Everyone knew when the annual get together was, and it could also be used to vote on politcal as well as military issues.
There were also "private armies" fielded by wealthy family consisting of their clients (a somewhat feudal feature) in the very early republic (pre EB). I imagine there was a similar practical procedure (get together, practise, the boss says go home or march out to war).
Carthage used mecernaries a lot, but apparently they adipted some Greek ideas from that Spartan chap (Xanthippus?) who makes an appearance in EB.
I beleive the Patricians were the old elite who had the right to hold priesthoods and public offcies like Praetor (the generals in the very early republic). I think initially all senators were Patrician? Anyway they had a strong hold on public offices under the Kings and early republic. A lot of their wealth was from land but they were not the only landholders, just the big ones. Over time some lost their wealth and some of the lower classes gained, which put pressure on the "status quo".
This led to their control being loosened by plebian agitation and "new rich" men looking for a role in the Republic. Eventually the roman class system became more open to money than blood. The "conflct of the orders " is pretty obscure as it was done and dusted in the 4th century, before the EB timeframe and almost all surviving histories so we get the rosy version "after a few frank discussions everyone agreed to some reasonable reforms, and that thing with the bloodshed was just a misunderstanding".
AFAIK any Roman citizen could hold land and the amount determined his level of military service and (along with his birth) his eligibility for public office including command. This goes on until the Principate, although military service became less inevitable.
Maybe the patricians are a bit like the "Ivy League" in the USA, you don't have to be one to make President/governor etc but boy it helps. They have advantages like wealth and sort of "unofficial" privileges (although patricians had exclusive access to some priesthoods that conferred political advantage and prestige: shades of Pontifex Skull and Bonesus).
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