I’m not a bike mechanic… but when I asked an engineer to get dead bearing out, a blind puller was what he used..
IME a blind puller is down right the most useful tool for doing jobs like this, once the inner race has fallen out. Admittely you can usually do without one if a bearing is in ‘good’ condition or can get to it (outer race) with a screw driver as you say. Stuff like cartridge wheels hubs are the most difficult IME, and where a blind puller comes in handy & a screw driver and hammer is not an option.
Not a cheap tool though, and when I need one the other day I needed 32mm+ size (which might be unusually large), which none of the ‘ebay’ blind puller kits cover as std (+32mm sizes are avaiable if you search the web).
The engineers at my old works sorted out a number of bearing for me with one of these, but for once in a blue moon job, it might be worth locating a local engineering firm or garage with one, rather than buy one yourself.