Does Anyone Find This 19th Century Style Count a Tad Embarrassing?

I know the first past the post system that most country's have is a lot easier to count.

Yes we do have a Proportional Representation single transferable vote multi seat constituencies whereas most other countries have first past the post single seat constituencies.

This is not correct. Most democratic nations have moved away from the plurality system (aka FPTP) and use some form or other of proportional representation.

It is only really common in former British colonies. Although admittedly the plurality system remains in use in two extremely populous nations: India and the U.S.

The main problem with lack of electronic voting is it means there is a flaw in the Proportional Representation System in the distribution of surpluses.

This is a serious flaw. But you don't need electronic voting to fix it. In Northern Ireland and Scotland they have a hand count but a much better method of transfers. We just haven't fixed the problem due to laziness.

There are however some further improvements to our voting system that actually would require electronic counting, such as the Meek method of transfers used in New Zealand. We would also save a lot of money by getting rid of the manual count.

The ideal first move would be to an electronic counting system using the current paper votes where they are scanned and counted. That would also solve the anomaly where surplus votes are randomly distributed.

I didn't think electronic voting meant actually voting electronically.
Surely we could have filled out the vote sheets as we did this time ,but computers read the papers just like mail centres sort out postal envelopes?

Exactly. We should use paper voting but electronic counting. Then the system would be protected from fraud by the paper trail.

If the paper ballots are present, what's to stop a candidate who is losing from demanding a manual verification of the ballots?

If candidates cannot demand such a verification, then we have to trust that the optical scanners and counting hardware/software haven't been tampered with, so having the paper trail doesn't really help.

If candidates can demand such a verification, isn't it likely that the losing candidate will always demand such, which means that we have to do the manual count anyway?

The best method would be to publish the content of every ballot paper online. Then anyone who wanted to could use their own software to verify the outcome of the election. I think it should be possible for tallymen and other observers present in the count centre to confirm that what's published online is accurate.

I'm not necessarily advocating optical scanners. What I consider most important is that the count is by computer, so that more advanced variants of PR-STV become practical.