The Church never denied the laity communion. For a long period of time in the church, however, it was common practice for the laity to refrain from receiving communion and therefore some priests presumed no one was coming forward for communion and did not distribute it. But this presumption is quite different from an actual refusal to distribute communion.
The Church usually celebrates under one species due to the fact that Christ is fully present under both species. To receive the host is to receive Christ whole and entire.
On a practical level, communion under only the species of the host is the least problematic method of distribution.
