Flow reactors always struck me as the distinction between chemistry and chemical engineering. Chemists mix up little batches in borosilicate glass. Chemical engineers use stainless steel pipes big enough to crawl through.
Think about it like this -- if you have a molecule flowing through a tube along with the molecules it reacts with, you can flow it at a controlled flow rate through a hot zone or catalyst bed and largely the "plug" will flow through as a unit and so the total time the mixture is at a particular condition is very consistent.
Whereas for a batch reactor, you basically have a giant cooking pot with a stirrer agitating the molecules together. But nothing is typically removing the product, so some of the product can get destroyed or be less efficiently made because the time of reaction for different molecules varies.
That said, a company I used to work for was doing batch chemistry at hundreds of liter scale reactor size. Some of these batch reactors are enormous. And they make sense for situations where you need to do something like slurry processing or super high pressures (sometimes) or are not at a big enough scale to justify building a flow system. For instance, if one 200L batch meets your production needs for a year there's absolutely no economical incentive to switch to a flow reactor at such a small production rate.
But in my experience, yes, I can make a flow reactor with yields of 99% for somewhat tricky chemistry. With a batch system I might hit 95%. That isn't a big deal for a high value product, but is a huge problem for a commodity chemical.
I'd say it's more about scale than continuous versus discrete reactions. Flow injection systems in a chemistry lab are table-top sized, whereas chemical engineers deal with industrial scale.
With flow chemistry, it’s not good enough to know just the chemical side of things. Other factors like mixing, residence time, different flow types, etc becomes much more important. That’s where a chemical engineer can really add value.