I have Shimano MTB shoes that are similar to the Tahos (which were the shoes I had previously), and both have the strap.
However, I have had ones without, and there are two ways of dealing with the laces -- thread the ends and loops down the first crossover or, thread them through the loop at the end of the tongue so they are away from the chain. I've never had a problem with laces getting caught in the chain with this method.
I've never used straps-only shoes. The pair of Specialized MTB shoes I tried before buying the Tahos were too tight across the top of my foot. And I am just a laces sort of guy.