I think the biggest value add - from a user perspective only - is as follows:

1.) accessing a local directory of restaurants with their menus

2.) accessing a common user interface experience for ordering

3.) not having to enter credit card info

4.) common portal to receive updates from the restaurant on the delivery

The actual dispatch of the driver and hiring of the driver is not a key component of the user experience and could easily be handled by the restaurant owner.

I feel like Square is uniquely positioned to provide 1-4 given their existing access to restaurants via Square POS and Square websites (Weebly).

Why would it be better if restaurant itself handles deliveries rather than a network of drivers?

I think its more that delivery specifics is a don't care for most end users - other than being able to see where the driver is. Actually the key value-add for the delivery apps is the credit card on file. Typing in 16-digits CCNs and 3 digit CCV and expiration for every restaurant every time you order (and perhaps more importantly, worrying about the safety of doing so) is a huge hurdle.

Yeah; I am asking though because it seems many here seem to say or at least imply that it would be better if restaurants had their own drivers, however I fail to see how it does not have overhead or could be more optimal.

So I suspect it stems from irrational hatred towards middleman services.

Is there enough demand for non-standard food types (I have to imagine more folks get pizza delivered, than say, sushi) for each restaurant to maintain their own group of drivers?

Not saying the end result has to look like uber eats, etc, but there do seem to be efficiencies from having X drivers collectively cover Y restaurants, with X << Y