Trichlor has CYA and Cal-Hypo has calcium. It's not just "in the tablets" but part of the chemical itself. The following are chemical facts independent of concentration of product or of pool size:
For every 10 ppm Free Chlorine (FC) added by Trichlor, it also increases Cyanuric Acid (CYA) by 6 ppm.
For every 10 ppm FC added by Dichlor, it also increases CYA by 9 ppm.
For every 10 ppm FC added by Cal-Hypo, it also increases Calcium Hardness (CH) by at least 7 ppm.
Some people do use Trichlor tabs if they are going away for a week if their CYA isn't too high already. If you don't have a pool cover and the chlorine demand is 2 ppm FC per day, then after one week the Trichlor tabs would add 8-1/2 ppm to CYA which may not be too bad. They are acidic, however, so you'd need to start with the pH a little higher and over that same week they would lower the Total Alkalinity (TA) by 7 ppm. So you could increase the TA somewhat before you go and have the pH start off at 7.7 or 7.8. You would then likely end up OK by the end of the week.
The reason there are no tablets that don't add either CYA or CH is that the chemistry does not allow for it. There is a powder/granular form of chlorine that doesn't add CYA or CH called lithium hypochlorite, but it is very expensive and doesn't form tablets (it dissolves too readily). Even Cal-Hypo tabs tend to dissolve too quickly and have extra "binders" to try and prevent that, but tend to leave residues as a result.