Are there transposons which create homozygous genotypes in offspring?
I study transposons for my PhD; I am not aware of any that preferentially insert through an active mechanism to an un-inserted allele.

There are tranposons such as R1/R2 which have a 'favorite' insertion site and will insert/compete for that sequence. That's kind of similar to your question but not generalizable.

The predominant factor I can think of which would cause such a thing to arise would be gene conversion in which the insert allele is used as a repair template for an un-inserted allele. Gene conversion is rare but can explain how the TE insertion is found on multiple haplotypes.

It's very unlikely that the alternative hypothesis occurs in which the same type of TE inserts at the exact same position in two different haplotypes and that TE doesn't have a preferred insertion site.