Australian ballots are not always simple, and you vote for your local representative so each district has a different one. Often you will have multiple ballots to complete, and some have the choice between above or below the line voting (where above you select some preferences and below you select all preferences, where all may be over 100 boxes).
Parties often hand out small leaflets outside the voting locations with instructions for above the line voting, which is typically a list of around 6 preferences in order.
Despite this, voting in Australia is done with paper, and we have live coverage with real-time counts on election night. Paper works quite well, computers really aren't necessary.
This is one of the very weird things about the US system, that somewhere in reconstruction the parties were effectively nationalised and allowed to use state-run election infrastructure.
The concept of "registered for party X" doesn't exist in European systems. The state has no business running internal party affairs, they can select their own candidates.
For example, in many areas the school district legally conducts separate elections based on separate precincts. But, no one is excited to take a separate trip to the polling place just for the school district election, which results in very low turnout. To improve turnout and save money, the school district questions are combined onto the same ballot as the municipal election - but two voters in the same municipal election precinct may be in different school district precincts. Imagine how this challenge multiplies as you add soil and water conservation districts, judicial districts, economic development areas, county issues, etc.
Last election I walked into my polling place six minutes before my bus was supposed to arrive right out front and caught my bus.
Skimping on costs when it comes to elections is pretty silly - it makes sense to make sure there isn't rampant corruption but people have enough reasons not to vote that adding inconvenience to the list can be quite discouraging.
In the US the ballot I get in my polling place depends on where I live and sometimes what party I am registered with. My ballot will likely be different than the next person in line. That ballot is many pages with numerous questions per page. And for about half of the questions I am permitted more than one selection.
Machines are a good aid here for this system. Verifiable paper tallies are good for election security though.