Yes you can buy & sell the same stock within a day
NO your broker does not need a few day for the stock to settle
You just don't understand the basics
Every broker/dealer in North American maintains their books and records on the settlement date basis, when there is a trade, it settles in three business days.
The rule is that if you purchase a security, it must be paid in full (or the proper margin is deposited) before you can sell it. If you sell it before making payment your account is restricted for 90 days or until payment is made in full or the margin call for the purchase has been made and you can not use the proceeds of the sale to cover the purchase.
If you are going to trade or even invest, one should always know that they are doing, why they are doing it and how to do it.
Therefore, when you purchase with the idea of trading, make sure there are sufficient monies in your account before making the purchase. The money on deposit must be good funds and in a sufficient amount to pay for the purchase in full or at the least to have the purchase properly margined. (Just be aware that before you trade in a margin account you have submitted the proper margin agreement and it has been approved by the B/D)
from the street