No Rome grow from city-state into an territorial empire. it would count if Rome remained tiny,still dominated Mediterranean, and forced the various states on the Mediterranean into becoming allies and providing Tribute.
What you are describing
is Rome. Or at least: that's how Rome began. The arrangement didn't last, because it
couldn't last. Vassals demand protection against the bothersome barbarians who raid them, and that protection is why they generally pay tribute without muttering, instead of fighting to the death against the hegemon's overlordship. When the hegemon subdues the barbarians,
they become vassals. They want the same deal. Thus, hegemony/empire is an ever-expanding thing. (Also: it's a wealth-pump, driving wealth from the periphery to the core-- which is why empires invariably start their decline once options for expansion run out.) Anyway, at the same time, the oldest vassals become culturally acclimated to the hegemon's culture, and they start to demand rights as full citizens. Which (after some trouble and revolts etc.) they invariably get.
Which is how hegemonies become ever more imperial, ending up as fully-gledged empires. This answers your question. Yes, it can happen, and in fact it did happen in OTL. But it can't and won't stay that way forever.