Dunno how things are done in the US, but ballot boxes are sealed here (with actual lead / hard to change seals). The boxes are then couriered (with several different people accompanying the box) to a central location. There are various different registers that show who has attended the vote, what papers have been used. ie. Double Entry. with different people responsible for each register. Usually with a completely separate observer overseeing the ballot box.
Lead is not hard to find and if security keys can be replicated from a photograph then a standard seal should not be much of a challenge. Who picks the people accompanying the box to the central location? Can the person picking them be trusted? Is a single team carrying a significant fraction of the ballot boxes? And if they constantly have people supervising the ballot boxes, how can they forget them at the polling station? And recounts don't always happen immediately (at least in the US) so the issue is not just transport, it's also storage. Who picked the storage area? Who has access to it? Is a team posted 24/24 to verify nobody enters that room? Who picked that team?
The room is sealed / guarded.
The room is sealed? Why? Do they want to prevent the general population from overseeing the counting?
It would take an amazing level of conspiracy and corruption to rig a count in the UK.
From what you've said I'd say on the contrary that all the conditions are met for tampering.
There are no volunteers, these people are usually paid (and paid well enough) for their role in the ballot and count.
Who picks these people?
Consequences for interfering with the vote in any way are harsh and will include criminal charges as well as most likely loss of employment (staff typically are Local Government staff).
The consequences for murder are even harsher. And yet that has never prevented them.
If you've ever been at a count or worked with the people at the polling stations you would understand.
I've been at a count many times but here it happens right at the polling station, as soon as voting is closed so that the ballot box never goes out of the voters control. The counting is done by teams of four voters who volunteered at the polling station during the election, in the open. The count also happens in the presence of party representatives of course and any one who wants to oversee it (which I've done many times too). If you wanted you could arrive in the morning (done that too, was first to vote), see the ballot box being prepared, see that the box is empty (it's transparent), gets locked with two padlocks using different keys, handed out to separate persons, and stay all day until the results are announced after the count. In other words anyone can control everything from the start to the end.
I totally agree that paper voting can be much more secure and reliable than electronic voting. And even with the flaws of the process you described it probably is (mass tampering would be harder for one). But 'paper' is not a magic bullet. It still has to be done right.