I'd reverse your order I think. Make a quota with 19 cells, with the logic for each set to Always Qualify. Then, add your blocks through the Block Randomization option in SSI Web, but use a constructed list to control the block.
For the lists, make a parent list with 19 list items, then make a constructed list where you put ADD(ParentListName,QuotaName). QuotaName will return the quota cell they were assigned to, so you would get something like this: ADD(ParentListName,4). The 4th list item will be added, which will have the respondent see block 4 and nothing else of the survey.