Does accessing WordPress via REST API affect the site stats and analytics?

This will depend on the type of plugin you use for counting page hits. In general stats are being counted on front end, either by inserting some image or JS which generates a server hit on the stats server that will be counted (google analytics, wordpress.com stats) or by counting the hit and storing it in the wordpress DB when the front end page is actually being generated.

The Rest API do not have easy means to add tracking JS and images, and do not follow the front end page generation code flow, so the answer to your question is 99.9% it will not be counted. Counting those types of requests will require some different strategy than the current plugins employ, but what is true for now might change in the future, although I assume that such tracking code will have different tracking for front page tracking and API tracking.

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error code: 523