There is currently no API feature to start a discovery scan. You can create scan job on a schedule to discover assets. Most customers would create weekly scan jobs to find host/assets and then add them to the Password Safe.

https://beyondtrustcorp.service-now.com/csm?id=kb_article_view&sysparm_article=KB0017147

If you are looking at automated task of adding assets to the BeyondInsight/Password Safe you can create REST API using our BeyondInsight or Password Safe API that will execute automatically during a server deployment to add the asset into the BeyondInsight/Password Safe. 

hi,

Also, how does other customers are handling (discover & onboard) the newly created servers in day to day?  

Diacover the assets that you intend to onboard.

https://docs.beyondtrust.com/bips/docs/bi-on-prem-discovery-scans

onbaord via smart rules and smart groups

  https://beyondtrustcorp.service-now.com/csm?sys_kb_id=09809e3a478ce21c1bf1db37536d43c0&id=kb_article_view&sysparm_rank=1&sysparm_tsqueryId=0436237b47e8aa18b77b3ddbd36d43d3       

                                                                                                                             https://docs.beyondtrust.com/bips/docs/bi-smart-rules

Periodicity of discovery? Daily/Weekly? And if weekly then how do you handle the new systems?   

adhoc discovery scans can be converted to scheduled discover scans that can onboard new assets

https://beyondtrustcorp.service-now.com/csm?id=kb_article_view&table=kb_knowledge&sys_kb_id=a7a3f34147978290b77b3ddbd36d438e&searchTerm=scheduled%20discovery%20scans                                                                                                                             https://docs.beyondtrust.com/bips/docs/bi-on-prem-discovery-scans#check-completed-and-scheduled-scans

Hope this helps.