You are right, instead if you work at all they have to pay you for the whole week.
It’s quite normal to see people start Tuesday on payroll, finish on Thursday, and they don’t get paid those whole weeks. They typically get those days paid, even if for example you “offboard” by lunchtime on your last day of work.
From California but similar everywhere:
> Pursuant to California law, an exempt employee must receive his or her full salary for any week in which the employee performs any work without regard to the number of days, or hours worked.
https://www.yourlegalcorner.com/articles.asp?id=135&cat=emp
If employers don't like this, they can pay overtime instead.
In this case with OP, my guess is that left them two choices:
1/ Accelerate start and then as you note, pay the whole period even though it sounded like this occurred before Christmas, was only a few calls then a vacation.
2/ Pay an invoice, which is mucky since it’ll probably involve supplier setup via AP, they’re clearly a W-2 equivalent person so perhaps this complicates other things like tax or benefits, etc.
If the person is “unencumbered” (not currently employed or subject to something contractual like garden leave which prevents them assuming the new employment) and if there was real value to having them participate in the calls then certainly the easiest would be (1.) - and IME, this is how larger employers would do it.
Consider this, I don't know what this software does but a lot of the conversations around software are confidential. Did they just invite a random stranger into a call with potentially privileged information?