Short answer
Bespoke software makes sense when a process is truly unique to your agency, or when the packaged option would force heavy configuration you will pay for every upgrade. It also makes sense when a smaller targeted tool can sit alongside your core ERP and do one job far better.
I think most governments should still run their core ledger, payroll and procurement on a proven platform, because those processes are governed by rules that do not change much from one county to the next and the big vendors have spent decades getting them right. Where bespoke software earns its place is at the edges, in things like a permitting workflow that reflects your local code, a grants tracker that matches how your federal awards are actually managed, or a dashboard your council will actually read, and AI has made building those faster and cheaper than it was even two years ago. What I would watch is ownership, so make sure your agency owns the code and the data and that someone on staff or a partner you trust can maintain it, since a bespoke tool nobody understands becomes its own kind of lock-in. If you want help sorting which of your processes belong in the package and which deserve something bespoke, the Start at Zero Navigator has a worksheet for exactly that.