Can the Fujifilm X-T1 do weddings?
For most – not all
Limitations for some may be the lack of a dedicated TTL flash and or a fast long lens. These limitations are only temporary though since a Nissin TTL flash is coming soon and more lenses from Fujifilm are as well. In the mean time manual flash works great as do most of the radio slaves on the market. Personally I have used an affordable LED light for occasions when window light is not enough or available, an old set of "Pocket Wizards" in the studio and a 300mm Nikkor with a Kiwi adapter when extra compression was called for.
The exception: the hardcore Pro
Less may be better
If outright AF performance is all you value, then a DSLR may be the only tool to satisfy your needs. On the other hand if you are finding yourself evolving into a more creative photographer and are looking for a lighter package that can still offer you the necessary performance without getting in the way then take a serious look at the Fuji X-T1.