Is your peer-reviewing service ever important for career advancement?
I do not believe simple refereeing activities have an impact on hiring or tenure, but it may have an positive effect in an annual report, to get “progress through the ranks” salary increases or bonuses, and other recognition at this level.  

It could have an impact in exceptional circumstances, such as receiving a referee-of-the-year- type recognition, but it would be truly rare for a junior person to get such recognition and thus have weight in a hiring decision.  For a more senior person repeatedly getting such recogniztion might be useful in a promotion file.

Doing a good job at peer-review is usually a stepping stone to getting an associate editorship or a senior referee position in journals, and these are often considered in tenure/promotion decision as evidence of oustanding reputation beyond the institution.

Note that outside public recognition of work done as referee or on an editorial board, rather than the anonymous work usually done as a regular referee, is usually what is prized.

I do not have a Publon account: I am tenured and consider it part of my job to contribute to my field by refereeing the work of others, just like I expect others to review my work.  I do get the occasional accolade, which I will gladly take, but  I don’t do referee work (papers or grants) for recognition any more than I give to charity for recognition.