Since not all points where one might need to show a ticket have access to the internet, it makes it hard to verify that the ticket hasn't been used yesterday/twice/etc.
Existing e-tickets mostly avoid this by being valid on only one service, preventing reuse.
I doubt that is true. Print-at-home issued in Germany, the Netherlands, Belgium and probably elsewhere are often "open" tickets valid on multiple trains.
There are various measures that could be taken to detect whether a print-at-home ticket has been illegally used multiple times with a good degree of success. For example, the ticket number can be checked with a central server at ticket barriers (where installed) and at on train inspections. The server will record the time and location of the check, and this information used to determine whether a ticket is still valid at later checks.
As you rightly point out, this will fail to detect multiple uses of a ticket when no connection to the central server is available. However, passengers have little control over when an inspection takes place, and a connection is probably available most of the time on most routes, so the risk of detection is still quite high.
Further measures could be taken to increase the chances of detection, even in these circumstances. When a connection is available, a list of used ticket numbers for tickets valid on that day could be downloaded by the inspectors' equipment. This would allow detection of a further significant proportion of reuse. The window available to use a ticket multiple times would become small - the time needed for a connection to become available and the data to be uploaded and downloaded again.
Also, where there are multiple inspectors on a train, the equipment used by the inspectors could communicate checked ticket numbers via, for example, on-train wifi. This would guard against the same ticket being used by multiple people on the same train.
In fact, I doubt the approach of checking each ticket against a database on a central server would work well for on-train inspections. It would often be quite slow, so a local store of information would likely be needed.
Other methods to reduce the opportunity for misuse include requiring a ticket to be presented with ID, or with the original payment card used to buy it. This would tie a ticket to an individual person, or at least one person at a time.
You can no doubt think of further ways to reduce misuse.
I'm not arguing that all abuse of print-at-home tickets can be prevented. However, I think a system could be developed that was secure enough to be widely used without significant losses. It would not be perfect, but no ticketing system is - including our current arrangements. Other new ticketing methods, such as smart cards, may be more secure and may offer even more convenience to passengers and the rail industry.