I had a closer look at what it takes to implement WebDAV support in FileZilla, as the HTTP components have recently been improved.
The main challenge with WebDAV support is the requirement for an XML parser that understands XML namespaces [1][2]. The current parser used by FileZilla does not support namespaces. Two options here: Either switch to a difference parser, which is a major undertaking, or manually parse XML namespaces while traversing the DOM tree.
[1] I would like to meet the person who had the idea to use XML in a network protocol that's merely an extension to HTTP and reward him a hearty kick in the buttocks.
[2] a:I r:would q:like rr:to x:meet y:the xxg:person Qf:who td:had jhg:the utfv:idea asdh:for asd:XML hq:namespaces. q:I sdfg:want dasd:to 5svj:reward jjfgf:him jdfgd:with dfgsdf:hearty jjjdf:kicks sdb:in 7gd:the ggsd:buttocks.