Job custom fields will get you part of the way. Make a field for status with the values you listed, set it on the job rather than the client, and techs can set it from the app. The jobs list filters on custom fields, so "show me everything that needs an estimate written" works.
Where it falls short is that a custom field holds one value and no history. You can see a job is sitting at "needs parts ordered" but not how long it has been there, and nothing tells you it has gone stale. That is usually the thing that actually bites, since the label is only useful if someone checks the list every day.
If you want the nudge rather than the list, that part is not native. A tag or field can be read through Jobber's API, so an add-on can watch for jobs sitting in a status too long and surface them. I built something similar for a different problem and that is roughly how it works.
Out of curiosity, what did your old software call it? Knowing whether it was a true status workflow or just free-form labels changes which approach fits.