While Twinkle keeps a userspace log of these forms of deletion nomination, the page curation toolbar does not. To maintain interoperability with twinkle, it should read the relevant section of twinkleoptions.js, if it exists, or simply check for the existence of the 'CSD log' and 'PROD log' user subpages.
I know that there is a deletion tag log: https://en.wikipedia.org/wiki/Special:Log?type=pagetriage-deletion&user=Insertcleverphrasehere&page=&wpdate=&tagfilter= but this log lumps PROD, CSD, and AfD taggings all together. Nobody wants to parse through all of that to check over their CSDs. Moreover it removes any ability to personally edit/organise your CSD log (adding comments/notes for future reference etc.) See mine for example: https://en.wikipedia.org/wiki/User:Insertcleverphrasehere/CSD_log
The other problem is that the Page Curation tools are currently unavailable on articles that are not in the NewPagesFeed (see task T207485 for a fix to that); because of this, articles that are CSDed from elsewhere are necessarily not contained in your page curation log, but have to be done via Twinkle and the userspace CSD log. This fragments the CSD logs into two different places, making the issue even worse. It is essential to be able to keep track of your CSD logs, as these articles are often recreated days, weeks, or even months later by COI editors, with similar problems (this is especially true of paid editing and SPI-banned accounts).
Requested multiple times at the Suggested Improvements page:
https://en.wikipedia.org/wiki/Wikipedia:Page_Curation/Suggested_improvements#40._deletion_logs
https://en.wikipedia.org/wiki/Wikipedia:Page_Curation/Suggested_improvements#55._Ability_to_log_CSD_taggings_to_their_CSD_logs
https://en.wikipedia.org/wiki/Wikipedia:Page_Curation/Suggested_improvements#64._CSD/PRODs:_update_userspace_logs