Personally speaking the plugin is only a shortcut to add the tracking code to your site and to give you an iframe to hubspot from the wp admin. It really does nothing more than that. Personally speaking I don't recommend any clients/organizations to use it for that reason as it is usually more efficient to run the tracking script from a tag management solutions (Adobe/Google Tag Manager) and then just log in to Hubspot for what you need there.
If you are on the free version for a trial and thinking about using Hubspot more for your business, you may be able to contact sales with this concern to see if they are able to investigate for you before you buy to see if this is just broken in the free version because the few times I do see this plugin used, it doesn't work and would not be related to your hosting.