The REST API allows WP to be compatible with a wide range of web services that go beyond browser based support. It makes WP “RESTful“. We recommend keeping it for maximum compatibility, in particular with mobile apps and possible future admin implementations. It’s very difficult to actually “delete” it, but it can be disabled.
By default, the REST API outputs a single line in your page head for discovery purposes. The overhead to do this itself is negligible. Loading the related code base to implement RESTfulness does involve some overhead, but compared to the entire WP code base, it’s only a small part, along the lines of less than 400kB of PHP source code. In any case, even if disabled, the code is still loaded. And fully deleting it in an otherwise compatible manner is just not possible.
Like most of WP, while the source code is almost always fully loaded, the vast majority of it never executes unless it is specifically needed, so the impact on normal HTTP browser requests is not that big of a deal, but it does contribute some. To know how much exactly would require some careful benchmarking. If you are that concerned with performance, you should seriously question why you are using WP at all. It’s not known for being highly fast and efficient. Still, if you have problems with site speed, the problem almost certainly lies else where, the core code is very rarely an issue. If you otherwise like WP, RESTfulness is not worth worrying about IMO.
Big thanks for your awesome help. I will keep it intact.