Thank you everyone for your patience and being so vocal with regards to this feature.
I'm happy to share that on June 29th we've released JIRA 7.4 which contains the HTML Export feature available to all customers.
As part of this feature, we've also introduced an ability to set limits to the number of exported issues so that you can control the load on your server. You can learn more in latest release notes: https://confluence.atlassian.com/jirasoftware/jira-software-7-4-x-release-notes-902079634.html
Thanks again for your help and feedback!
Product Manager, JIRA Server