Several things to keep in mind. What is the nature of the advertisement?
Is it (A) an actual position which you have to apply to through either a University hiring portal, or (B) by initiating contact with an application package to the PI... or is it (C) a broad statement on their website saying interested postdocs should contact the PI?
If (C), then you should not submit any formal cover letter but rather a brief professional email with a CV and 1 or 2 representative papers attached. In that email you should mention you are looking for a postdoc, a short statement that you are interested in the research area the PI is working on and leave it with an open question as to whether they have any upcoming postdoctoral positions, and if so, if they want to discuss further.
If (A) or (B), submit whatever application package is requested to the appropriate email/application portal. In your cover letter provide your brief introduction and statement of purpose.
In my opinion, there is no need to discuss at length a particular paper, and certainly no reason to discuss what you think are "improvements" to that paper to make it more "meaningful". You can dig yourself a hole in many ways; the PI is no longer working on that project; your suggested improvements are wrong; you come off looking as if you think you know better than the PI and their team, etc... besides the paper is published, it is what it is. Once you get an interview or are hired, and assigned a project, if appropriate, you can bring up potential follow up works which may improve that line of research.
The appropriate discussion is you addressing that you find the work interesting - please NO PRAISE of the PI (I know this is cultural, but in the US a letter with lots of praise about how esteemed my research group is.... is just awkward). Then say you have read some recent papers, you can list 2-3, and say you think you could both contribute to this general line of work and grow your expertise and career by joining the group.
Then list your relevant skills, in brief, and where you can see them fitting in with the group. Here you can identify briefly something you can add to previous work which has been lacking... e.g. if you are good with theory and the prior publications lacked it, you can say that is something you bring to the group. This is different than insisting that the specific prior work was less meaningful without theory, but rather that it is an extra dimension that makes you stand out.
