Same in my country, never thought of this as being outdated, just the simplest most cost effective solution for a system that has to be totally secure.

This is literally how the Dutch election system has worked for years, ever since they stopped with the voting machines (which were admittedly a mistake).

The recent change has to do with how the votes were tallied. Basically the way they were transmitted wasn't really tamper-proof, and there wasn't a system in place to determine the difference between the result of the ballot station and the publicly announced result.

Of course they've 'solved' this by replacing the electronic tallying system. Which I can't help but feel is a very misguided way of fixing those problems.

This isn't about that step, as that is done by hand in the Netherlands, too.

It is about what happens after that. The electronic steps after that are deemed not secure enough.

How are the results transmitted and tabulated by hand?

Not the OP, but my country runs a similar system. Each election station counts the ballots by hand. Anyone can signup for counting and/or watch the counting. It's mostly reps from all participating parties. Sometimes few independent citizens show up as well. After all people in the station agree on their total count (they can recount if needed), they phone in their numbers to the central station.

Raw numbers of each station are available publicly. The people who did the counting can check if their numbers were transmitted correctly. Parties usually have preliminary results from their people at stations way before the centralised system processes & publicises them. They'd easily spot any issues. Paper ballots are stored in secured bags and can be recounted if there is a need.

Is there no point (e.g. during transit) when ballots are out of sight and could be intercepted/replaced?

Watching the BREXIT vote counting process was similarly heartening.