Short answer
Most governments do better with a phased approach, usually core financials first and HR and payroll next, because each phase is smaller, easier to test and easier for staff to absorb. A single big go-live can work when your current system is truly at its end and your team has the capacity, but it raises the stakes on every decision.
I think the choice comes down to risk and capacity rather than ideology, and the agencies that do this well are honest about how many hours their finance, HR and payroll people can actually give the project while still closing the books and paying employees. A phased plan lets you learn from the first go-live, fix how you work with your implementation partner, and carry those lessons into the second phase, which in my experience makes the second phase noticeably smoother than the first.
There are good reasons to go all at once, like an old system that is losing vendor support, integrations between modules that would be awkward to maintain temporarily, or a leadership team that wants one disruption instead of two. If you go that route, I would invest heavily in testing, parallel payroll and data conversion rehearsals, and I would make sure your contract pays for those activities rather than leaving them as "agency responsibilities" that nobody has budgeted for.
Whichever way you go, the key to success is sequencing around your calendar, so plan go-lives away from fiscal year end, budget season and open enrollment, and build the interim interfaces into the plan from the start. That way the phases feel like a series of wins your council can see rather than one long wait for a single date.