My Parish also offers only the Host at weekday Masses, other than Holy Days. The main reason we only offer the Chalice with the Precious Blood on the weekend Masses is the lack of Eucharistic Ministers, especially at early Masses for those who work during the week. These laity who serve as EMHC (Eucharistic Ministers of Holy Communion) must arrive early, and if there is no Sacristan every day, must also remain and assist in setting up the Altar for the next day as well as replacing Hymnals, Missalettes, etc… Some of them must also get to work, or have children at home and can’t arrive at 6:30 or 7:00 am for the weekday Mass, nor can many of them remain if there is a second Mass later in the morning. It usually requires two of them, so the members can divide in two directions after receiving the Host from the Priest and go to one side or the other. If there is any of the Precious Blood left, the Priest and/or the EMHC’s must consume what is left. (And this is before eating any breakfast!).
Therefore, we only have the Host during the week, but offer the Eucharist under both species at the weekend Masses. Since I do the scheduling for one or two months at a time at one of our three Parishes on the weekends, I KNOW how difficult it can be to be certain both EMHC’s are present at the week-end Mass! I often have to fill in for those who “forget” to come, or decide to take a week-end trip without warning me ahead!
As for having both species in a large Cathedral, this may be due to the crowd. They may often (as even our small parish does) use all of the Consecrated wine before even 1/2 of the members attending get through the line, and then there are complaints from the rest wondering why we don’t use a much larger Chalice! (We simply can’t put a quart or a fifth of wine in each Chalice and have it be at all reasonable to handle for large crowds, not to mention the expense involved for the Parish). That’s probably why you don’t see it at much larger Cathedrals on the weekend. Smaller Parishes may offer both on the weekends. Try it.