The shortDOI service will work for referring your readers to the source, but is not widely used, and its use is even discouraged behind the scenes. As of July 2022 the shortDOI site states it was updated in April 2013, which gives some indication of its current status. The problem with shortDOI is that it just provides an alias for (i.e. link to) the actual DOI, that has to be created and maintained on top of the existing service, and then transformed whenever used. This made sense for its original purpose as a link shortener for, e.g., emails and mobile messaging, but with other uses it results in duplicate paper identifiers that cause extra work for anyone maintaining indices of papers, for example. And since shortDOIs never got that popular in the first place, not all APIs/services support them.
Journals and other publication venues are recommended to not use shortDOIs, but to make shorter regular DOIs. See this Twitter thread or Crossref's DOI display guidelines (emphasis mine):
The International DOI Foundation created the ShortDOI service as an open system that creates shortcuts to DOIs. DOIs can be long, so this service aimed to to the same thing as URL shortening services. For example, for the DOI
https://doi.org/10.7774/cevr.2016.5.1.1, the short form is
https://doi.org/bwfc. Only one ShortDOI is created for each DOI, and the ShortDOI returns exactly the same results as the DOI. ShortDOIs are not widely used and are not really actual DOIs themselves, which is confusing. We recommend simply creating shorter DOIs in the first place. Learn more about constructing your DOIs.
Given the above, I'd suggest it's generally better practice to use the regular DOI in bibliographies. This is especially the case if the DOI is only used as a hyperlink rather written out in the bibliography. That said, I am not aware of any journal publicly taking this position. Few journals seem to mention shortDOI at all, but there are some that make its use optional. For example, Workplace Health & Safety's author instructions state
When a DOI or URL is long or complex, you may use shortDOIs or shortened URLs if desired. Use the shortDOI service provided by the International DOI Foundation (http://shortdoi.org/) to create shortDOIs.
I was also able to find one journal that requests authors using shortDOIs: Cognitive Sustainability. Their Submission Preparation Checklist includes
Where available, URLs and DOIs for the references have been provided. Please use ShortDOI (http://shortdoi.org/).