It's not extremely difficult to gather logs even if the user clear the logs or use incognito mode. You just do it on the network level, or a proxy. Typically you wouldn't get the details about the requests and the payload from the response (because it's encrypted using https), but if you require going through a proxy to reach the internet, then you could log it all. (It's been ages since I worked in this field so I might be wrong on some details here and there.) Then it's just a matter of knowing who your customer actually is. It sounds like a difficult sell to require customers to show a valid ID to be able to pay for their computer time.
Overall it sounds like a business that I wouldn't go into: overly complicated, not very interesting for the customer, and little to no profit.