It would be helpful if you could provide a source supporting your thought that an ETF following a particular index would buy stock in the new company over what you describe as an extended period of time - as opposed to within a short few days.
I spent quite a lot of time before the SpaceX IPO learning how a change in Nasdaq 100 rules regarding adding a new company to its index, would impact ETFs whose prospectus tied them to the Nasdaq 100. The information I reviewed indicated the ETFs would add all of the new purchases immediately (within a matter of days following the addition of a new company to the tracking index).
I spent quite a lot of time before the SpaceX IPO learning how a change in Nasdaq 100 rules regarding adding a new company to its index, would impact ETFs whose prospectus tied them to the Nasdaq 100. The information I reviewed indicated the ETFs would add all of the new purchases immediately (within a matter of days following the addition of a new company to the tracking index).