Part of the interview process is selling the organization and its people. Maybe companies prioritize phone screenings and interviews first to steer the process and educate the candidate about the company and the position. Maybe candidates would be less likely to try the coding project if they hadn't built some rapport with an employee.
> They should always do the take home test first or after a short phone call.
Absolutely not. If most applicants need to do 4-8 hours worth of homework, they'll never have the time to review the results for every applicant. And making me do 4-8 hours of work without getting it reviewed thoroughly and getting personal feedback is not valuing my time.
In the process above there already was a screen sharing coding test. Adding the homework on top is redundant. You should be able to tell if someone can code in one hour of screen sharing, but you can't really tell if someone is mediocre or top notch from a single homework assignment.
Only once have I been given a homework from an interview and it was a solid 10-20 hours worth of work. I refused to do the work and I got the offer anyway, but I turned it down in favor of another offer.