By request? No. It's not up to the customer to stipulate the remedy - just read the warranty. That's the document that, among other things, explains the remedies available to the customer. As in any warranty, from any company, no matter how "premium" the product, the company, not the customer, gets to determine the remedy.
Whole-unit replacement is always a possibility, based on Apple's decision. Traditionally, if they're not able to do the display replacement in a reasonable amount of time, they may choose to offer a replacement. However, on a day-to-day basis, a display replacement will be Apple's first choice, because overall it's more efficient and cost-effective.
Should I bore you with the details? First off, if whole-unit replacement was the standard approach, every store would have to have sufficient inventory of replacement units matching every configuration. Specifically in this case, every amount of Flash storage, but depending on the model, every cellular radio type (GSM or CDMA), too (color is a variable in nearly every case). Since the same display module is used regardless of Flash storage, only two parts need be maintained in stock (space gray or silver display) instead of four (space gray 64 GB, space gray 256 GB, silver 64 GB, silver 256 GB). For models that come in more colors and more Flash configurations, it gets even worse (typically, even when there are three or more colors of case, only two display module colors are typically required - white or black, based on the bezel color).
You'd likely think it'd be stupid for a car company to issue replacement vehicles instead of repair a defective part. Why is it so much harder for people to understand the same principle applies to iPhones, too? In the case of iPhones there aren't a lot of parts that can be economically repaired on site, but when those parts can be replaced efficiently, they're going to be.
Based on the comments of many others in this thread, who feel that the more expensive solution is preferable... This is Economics 101. The estimated cost of warranty service is built into the price of the product, just like the cost of physical components, labor, shipping, advertising, etc. The more expensive warranty service is (due to either defect rates or cost of repair), the higher the initial product price must be. In other words, if, as a matter of course, brand-new units are provided as warranty replacements (even when module/part replacement is all that's necessary), that higher cost must be factored into the original price of the product. As always, the customer will be happier to get more at the same price - who wouldn't be? However, since that would require the company to take a lower profit, and the only motivation any company has to be in business is to make a profit... what are the chances?
“Reasonable amount of time” as defined by Apple, not the customer. The six hours of driving to get to and return from an Apple store is already an unreasonable amount of time. Making the customer wait at the store while they try and fix apples mistake is insult to injury. But let’s go with the car analogy. I drop my car off for repair and they give me a loaner. When *I* am ready to pick it up I come back and get it and give them their car back. They made a manufacturing mistake and their job is to inconvience the customer as little as possible. They screwed up, and they are in no way owed to profit of the prior transaction.