If this is such a serious concern, you should automate this process as much as possible. You don't necessarily need to respond manually to these requests if you put in place the required features on your website which will allow your EU data subjects to benefit from their rights.
Realistically, how hard is it to automatically grab some data from a database and export it as JSON, as well as remove data from your database pertaining to a user? With a relational database, this would be a cinch. I mention the right to access the erasure right, as I estimate these will be the most frequently called upon.