Email Sent Activity Sync to SFDC with Lightning

We have Activities Sync on for "Email Sent", which creates an email activity in Salesforce. This email activity record in Salesforce doesn't have the content of the email itself; it instead has the link to a Lightning email template, which further takes SFDC users to a Classic email template. Since our org already moved to Lightning and most of sales people don't have access to Classic email template, passing these "email sent" activities don't show users anything but the subject line of the email. 

I also noticed that, since the link within the synced email sent activity takes us to the template, token and other personalization parts are not populated for each lead/contact. As we're trying to expand our use of dynamic content etc, this is a concern that our sales folks won't be able to get intel of what exact message a lead/contact has received. 

Does anyone has any idea how we could pass the actual email content, not a template, into the SFDC email sent activity? We're not using MSI as it's cost-prohibitive.