Look at a red hawthorn, or perhaps a crab apple where you get both flowers and decorative fruit - there are many lovely varieties to choose from, and you can make jelly with the little apples if you don't want to leave them for the birds.

It is a big job removing the roots of a chestnut - you will need to get them out if you want to plant in the same place or growth of the new tree will be affected.

Fast growing usually means a lot bigger than 10 metres eventually - have a look at Betula utilis Jacquemontii 'Doorenbos' which gets to about 15 metres eventually, it's a rapid grower, but, if grown as a multi stemmed tree rather than a single stem, will make about 12m.

I would also go for a birch tree, wonderful canopy shape. There are weeping forms and, though I don't know it, I suspect that 'Doorenbos' could have a white trunk?

Do watch out for the right variety though - some birches get very tall.

If I had the space for a chestnut then my ideal replacement would be a magnolia grandiflora - not fast growing maybe but I would sacrifice that for the sheer gloriousness of an unconfined magnolia.

Paulownia tomentosa or Catalpa bignonoides

Bulbaholic - yes, 'Doorenbos' is the whitest, looks fantastic with multiple trunks, rather better than with a single trunk I think.

Everyone has their favourites and mine would be amelanchier. Lots of varieties that provide white or pinky flowers followed by black fruits and leaves with late seasonal interest. Also as already suggested, Crataegus Laevigata Pauls Scarlet, double crimson flowering hawthorn tree. Wonderful flowers in May and attractive fruit in the autumn, does well on heavy soil.

Catalpa get bigger than 10 metres though, and wide with it.
Lovely tree though.