Various people accepted the indignant line being presented in some quarters about the renovations to Buckingham Palace. The Metro slants the story here, for example. This article gives a perspective that they may not have been aware of.
My feeling, for what it's worth ... I prefer the monarchy to most of the rest of the political establishment, as I think they have a long view, even if their powers are limited (I wish the whole UK government had a long view and limited powers....!). I think that Buckingham Palace represents a national asset, which a "lucky" family gets to live in, in return for surrendering pretty much their whole life to the country. There aren't many 90 and 95 year olds who aren't really given the option of just retiring. Buckingham Palace is part of the heritage of the country which brings in a lot of foreign money. The lack of renovation for 60 years (! Do you have a 60 year old boiler in your house? 60 year old wiring?) is poor form, and I think that to see the palace go up in flames for lack of proper maintenance would ... well, be oddly fitting in a country which is burning most of the rest of its things of value, but would nonetheless be a disgrace.
Furthermore, I think that even were it the government funding this, there is no way that the money would be directed instead in directions that we think it ought to go - that's not the alternative.
A relevant comparison - the refurbishment of the Houses of Parliament is estimated to cost around ten times this amount (estimates of cost vary between £2 billion and £7 billion pounds), and this will be funded by the taxpayer. And government departments don't have a good track record of securing value for UK taxpayers' money.
The angle on "doubling the funding for the Royal Family" is an interesting one. The crown currently pays tax of 85% on its income, which leaves a balance of about £40 million from which the Royal Family and their estates are funded. So if the Royal Family weren't there, the UK would have £230 million less income. For the next ten years, the amount of money taken from the crown's income will be decreased to 75%, to fund this refurbishment. It is true that the UK will have £370 million pounds less money over 10 years. But the sotto voce implication is that without the crown, we could use that money ourselves. In fact, over the course of that ten years, even with this refurbishment being paid for, the UK will expect to receive an income of £2 billion pounds from the crown.