I don't use or carry a phone. The long and short of it boils down to things working just the way they do with a phone, just more human involvement, like requesting a paper menu, or handing someone money directly. You make plans to be at a place at a time, then you just show up.

> bank

Choosing your bank according to the provision of acceptable services.

> messengers

You just need an OS somewhere (not necessarily a smartphone)

> travelling

Cannot really help: if some administration requested a smartphone, I would either try to avoid it or buy some provisional, temporary thing.

Note really an issue, I've never had one. Banking is mostly on paper, sometimes I phone them up. Not bothered with messengers, travelling in the EU, there was always a paper-form alternative for that. The main upside is the looks of disbelief (and occasional panic) when you respond "don't have one" to a demand for mobile number :-)