Anecdotally, almost all small online businesses I've seen the back end of do not have a dedicated logistics person. When there's only a handful of different kinds of stuff to ship, shipping becomes an extra low-skill task usually given to whoever is free. They also usually do not have a sophisticated inventory management system (because it is not worth it to purchase one), and instead use spreadsheets that are manually managed.
Secondly, sending out stuff same-day is not even close to same-day _delivery_. For that you need to control all aspects of the logistics chain, which only makes sense at Amazon scales. I've encountered many times where I've ordered things, had a label created the same day but then with no progress for a week. After contacting the seller, it turned out the courier only picks up packages from their warehouse once every 3 days, and then takes another couple of days to "ingest" it and do the origin scan.
Back when we only had a couple of dozen SKUs, it was much easier to get everything out same day. Package them all in the same envelope, stick in a red post box. Done.