Haskell, 71 64 51 bytes
g#l=[[x|x<-[1..],c<-w,x==sum[1|e<-g,e==c]]!!0|w<-l]- Massively outgolfed by @pxeger answer, check it out.
Full list comprehension solution
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g#w=sum.f<$>((\y->[1|l<-g,l==y])<$>)<$>wf(h:t)|h<f t=h|1>0=f tf[]=[2]g#w takes g grid as a string with newlines and w words as a list of words.
Each word is transformed into a list of lists of 1's by g before using f to select the shortest(lexicographically) character.
f has an edge case = [2] for the end of the list which is always greater.
Then sum each selection to obtain the output