Well, what's a monarch? What does a monarch represent? How is a constitutional monarchy different from a republic?
I think the answer to your question lies in the answer to these questions.
Edit: a monarch represents hereditary, non-elected governmental power and authority. In a democracy, the power is vested in the people, not in a king or queen. Monarchy is fundamentally anti-democratic. A symbolic monarchy may only be symbolically anti-democratic, but the symbolism is still negative.
If you choose to see it that way
Wait... so maintaining a symbolic monarchy is like putting a vanquished enemy on display? In that case, wouldn't a cage be more appropriate to house the monarch than Buckingham Palace? At the very least, I'd think that if that was the actual attitude, people would consider it inappropriate to fly Queen Elizabeth around the world as a representative of her nation.