Would You Issue Public Bids If You Were a Federal Buyer?

Absolutely not, if you could avoid it! Why:

1. Public bids are inordinately expensive and the amount of time it takes to acquire the good or service in question is ridiculously long. 
2. They consume impractical amounts of staff time, assuming staff is available (which is rare). 
3. Because public bids are so visible, they result in increased vendor scrutiny, debriefings, complaints, and protests. 
4. Public bids, by design, result in the filing of multiple proposals! This adds to more evaluation work, more material to read, more material to score... you get the gist.
5. They are responsible for the death of many trees.

Why do public bids exist?
For the sake of appearances, political reasons and, in the case of large projects involving complex solutions, they are often in the best interest of the American taxpayer. 

Multiple Award Contracts (MACs) are rapidly becoming the buying mechanism of choice, at least until the system is reformed (maybe by 2050).  Also called Indefinite Delivery/Indefinite Quantity (IDIQ) contracts, multiple award contracts are awarded to an unlimited number of vendors.  Federal buyers then purchase from the list of awarded vendors, sometimes referred to as the "anointed" by outsiders trying to get in. Competition for the task orders issued to multiple award contract holders ranges from none to fierce and everything in between.