I've used software-as-a-service modules before to test this. This is what I use because I worked on/with these modules a lot so it's the most efficient route for me, but I don't think it would take anyone more than an hour to set up.
(This is a clone-based approach.)
Go to your hostmaster site and download the important modules:
drush dl hosting_variables hosting_saas
git clone --branch no-specifics https://github.com/PraxisLabs/hosting_restapi.git
Enable the modules, and set the site nid and platform nid in Hosting -> SaaS. (You can find the nid of a site or platform in the URL of their "edit" tab.)
Test the site creation with this bash command:
curl --data "url=test1.example.com&invoice=1" http://yourhostmasterurl/hosting/api/site
If this works, from there you can easily create 1000 sites with a simple bash script:
curl --data "url=test$i.example.com&invoice=$i" http://yourhostmasterurl/hosting/api/site
This would queue up clone creation for test2.example to test1000.example.com
Here's an article I wrote on using Aegir SaaS in production.
Alternatively, you can set up Hosting Services and use the REST API to queue up site creation tasks. I think this could take a few hours if you've never used it before.