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So, (how) would it be possible to get something like this as standard trajectory for politician? Obviously you can't have carbon copy of Roman one since offices don't exist and mixing politics and military wouldn't work.
But something like this: before you are elected PM/president you need to serve as minister. Before you are appointed minsiter you need to serve in representative body. Before you are elected in representative body you need to serve in local government (which is also required if you want to serve as major).
POD of your choice but world should resemble OTL as much as possible. so no "surviving Rome". and obviously pwoer of nobility has to be curtailed