"The Glow library is well designed, but why on earth did the Glow developers want to reinvent the wheel? The reason they give -- to support old browsers not covered by existing libraries -- just doesn't make sense."

Real reason :- They are spending their time and their money and the best people to decide how to spend both are the people who "own" both the time and money.

I guessSome devs managed to convince a manager(with budgeting authority) that this was a good idea.

Not that that's a bad thing. In general, I am all in favour of people reinventing wheels sometimes. Every reinvented wheel often fits local circimstances better, (omits unneeded functionality/generality etc) and is a good learning experience, if nothing else. Monoculture sucks.

So "why did they do it ? " is largely an exercise in rhetoric ... except for your key point ( I upvoted you btw)

"But at a time when the BBC is under tight scrutiny and intense political and budgetary pressure, why waste precious resources trying to compete when others will do it for free?"

The fact that they spent public money to do this is far more dubious. That said, large organizations waste a tonne of money on utterly trivial things - a few programmers salary for a couple of years (or whatever the time frame was) is likely to be small beans, relatively.