Tree looks lovely as it is. Does upset me when people want to butcher a perfectly healthy tree because of;
Seeds
Leaf litter
Shade
Stops you from drying washing - really?? Thought the wind would do that not the tree!
Stops the children from enjoying the sun - really - do you not ever let them out of the garden?? There is probably a suitable park/beach/river/open space near your home?
If that tree were TPO'd and you had made an application for those reasons you would have had a very negative reply!
Let's put this into perspective shall we;
Seeds - fair enough the average Birch will distribute 5,900,000 seeds per kilogram of seeds during a period lasting probably, and no more, 30 days. You still have 335 seed free days a year. Replant with a Sweet Chestnut and you'd only get 250 seeds per kilo and you could eat them at Christmas.
Leaf litter - the tree will generally shed its leaves around September/October time over a period of about 40 to 50 days and not all at once. Bearing in mind the leaves are quite small in comparison with a Horse Chestnut for example you are very lucky. The leaf litter acts as a renutrifying agent in the soil of your beautiful garden and saves you money with having to buy compost or grass enhancing mix. Also worms will be attracted to the garden and therefore aerating the soil reducing the risk of hydrology imbalance and flooding. Leaves will also remove the impurities in the atmosphere providing cleaner air for your children to breath. They also balance the thermal properties of the surrounding structures and regulate humidity keeping you and your family cool in the sunshine that causes cancer, dehydrates you and can kill in extreme conditions (we had extreme water so how about extreme hot like the Australians experienced this year?) Dappled shade of Brich v complete darkness under a Beech?
Shade from the sun your children are deprived of - lets be honest how many days sun do we get a year? For those infrequent days it is with us wouldn't it be more appropriate to take your children out of the garden and to the beach or somewhere equally tranquil to enjoy the pleasures of this fine country? Also the shade is transient, that big orange balloon that occasionally appears in the sky stays still but our little planet spins around it at approximately 1037.5646 mph (at the equator) and so the tree casts shade across different parts of your property. Move your deckchair to meet the sun?
Other elements you have not suggested;
Oxygen - the tree is providing you with much needed oxygen reserves. A fully grown Beech will sustain a population of only 10 people for one year.
Water - the average mature tree drinks 17000 litres a year - your garden is going to get soggy!
So in essence, that tree has probably been there before the estate was built or not long after. You moved in knowing the tree was present and did not take steps (like find another house south facing with no trees in view) and therefore taking it out on a poor defenseless tree is a crying shame!
Have another think about it!