Why are weather kites no longer used (much) operationally?
Most data we collect about weather is done regularly and for all altitudes of the troposphere.  Kites, though, are at the mercy of the weather and do not fly high enough.  So when there is ice or snow or other meteorological conditions that don't allow a kite to fly, you get no data.  We now use methods to measure the weather that don't break in bad weather.  The answer to your question is succinctly put in this NOAA webpage where they state these difficulties with kites: 


The average altitude reached was only about 3 km.
Data could not be evaluated until after the kite was reeled in...
Observations could only be taken in good weather with winds neither too light or too strong.
There was danger of the kite breaking away and endangering lives and property.


In the 1940s, weather balloons were being released and could gather an entire profile of data in all weather conditions.  This is far superior to a kite.  That being said, some people do still use kites: High-Flying Science, with Strings Attached.