Does a RESTful API in Caché remove the requirement to use a DLL and generate proxy classes for .NET web development?

very short:

.NET binding in Caché projects Caché Object classes to .NET with properties and methods
and services them with a private protocol. It is dependent on synchronized objects at both ends.

you may know this picture:

http://docs.intersystems.com/latest/csp/docbook/DocBook.UI.Page.cls?KEY=GBMP_intro

A RESTful API does something total different.

You define request (for data or actions)  and sent the request to Caché.
And you get it back preferred in JSON style expressed.
Your requests are straight HTTP with someWebServer in between on your way to Caché.

SOAP does something similar. But it has an incredible overhead of XML.
And as with .NET binding you need predefined data structures to be serviced.

In extrem:  With REST you just get back a string / stream and it is up to you to decide what this means:
No predefinded structures. It is all your task to organize yourself.
For .NET end I found this link:
https://docs.microsoft.com/en-us/aspnet/web-api/overview/advanced/calling-a-web-api-from-a-net-client
 

For Caché side there are lots of useful articles in the forum already and examples in SAMPLES.

Simple summary:

For RESTapi you will not need 
InterSystems.Data.CacheClient.dll nor generated proxy classes using the Caché Object Binding Wizard for .NET
You are the owner of the exchanged data structures.