As explained in Taamei Haminhagim #950, the minhag of bedeken comes from the Pasuk in Chayei Sarah:
"And she said to the servant, ‘Who is that man walking in the field towards us?’ And the servant said, ‘He is my master.’ And she took the veil and covered herself.”
If the basis for the minhag of bedeken comes from the verse, "And she took the veil and covered herself", then why is the Chasan the one who veils the Kallah? Why doesn't the Kallah veil herself? Wouldn't the minhag follow the Pasuk better if this were the case?