The tradition view
Saturday is linked with Shani, so some families avoid it for first entry into a new home. Others are comfortable if the family priest selects a suitable time. This is why you should ask the priest about entry, not just transport.
The practical moving view
Saturday is popular because people can unpack on Sunday. Movers know that. Discounts are usually weaker and late crew risk is higher, especially at month-end.
The middle path
Move heavy goods on Saturday if you must, then keep the first ritual entry for the chosen muhurat. Write the arrival window, crew count and lift slot in the quote.