We pay 325 for a day's work for two men. They do nine hours inc lunch. So approx 20 an hour each.
However this is more than a day's work for two men. As it is rotten then they cannot scale it. It's too dangerous. They will need a cherry picker or equivalent.
The issue is then once it's down. It takes a lot of time to remove the branches, and log it.
We also have a huge rotten one, and have decided to let it come down into the neighbouring field of its own accord. As the cost to get men into fell it will be way higher. We will then have to pay for it to be removed.
If you can let it come down naturally, if it won't damage any buildings, then consider doing so, as it will reduced the costs.
First off get someone out to assess. If it's stable enough they will abseil it, as per the pp, but if it is rotten and there is a risk it will come down with a man on it, then they won't do it.
Do you have a wood Burner? Can you keep the wood? Wood is very expensive and if you can season your own then it can offset the cost. If you don't maybe ask them if they will reduce the price if they keep the wood as they will be able to sell it on.
Willows are not legacy trees unfortunately. I've just planted one. As they are beautiful, but they live on average about 30 years.
The other thing to look out for is is it close to thr house? Willows are shallow rooted, and their roots are hugely invasive, so will go out for at least a hundred feet. They take a lot of water from the ground. You need someone experienced to check the roots have not invaded the foundations of the property as it could cause structural issues when removed as then there is nothing then removing the water any more and effectively the ground swells.