Using Zotero's style editor I undertook a try to adapt Chicago 17th to my local needs. 'title' variable has bee set to value 'none'. Style heading adapted accordingly. In the end I can't observe removal of source title in citation footnote.
UPDATE Well I see, this is no failure what I get currently in document, rather it is like Chicago 17th is expected to work. Very first reference in text to given autor gets full reference description in footnote. Only subsequent ones can be formed to short, name and date. One short test made and I got confirmation.

Thanks for hint. I take it in my serious consideration as it provides me with wanted result. Only that because it is paper in discipline of technology - no idea if ok if styles from other science branches are used. Must clarify with my supervisor.