No, it won't. Older printers would do this, but the HP found they could force you to buy more color cartridges by removing this feature.

Printers are relatively inexpensive to manufacture and last a long time before having to be replaced. Therefor profit for selling "expensive" printers is relatively low. You purchase an expensive printer and keep using it for 10 years and the manufacturer makes minimum profit.
Ink Cartridges are very inexpensive to manufacture and are relatively quickly used up. There is almost no ink in them. The cost is mainly in the circuitry and that is likely $10 or less. Therefore, high cost ink makes sense. Sell the printer cheap. Sell the cartridges for a lot of money. Make a minor profit on the printer (or sell at a minor loss) but make a 50-75% profit on each cartridge. HP8100 Pro ink cartridges at $45 for 700 pages plus $40 for 700 pages of black means $85 for every 700 pages printed. A business could easily go through 700 pages in a day. Best of all if printing only black and white the machine won't run without the color cartridges in it. If any cartridge runs out the machine won't run. You have to purchase a 3-pack of color cartridges even if you need only one color. Profit is increased if you turn in your empty cartridges as most are reusable with just cheap ink added or by re-using the electronics on a new plastic ink case. You get a minor discount. Note that I have NO factual information on the costs of personnel, equipment, advertising, shipping, material for any printer made. Sort of like with the drug industry. The printer companies may have to charge an arm and a leg for ink cartridges to stay in business, but I doubt it.
So, to answer your question in one word. Probably Profit.
At least this is my evaluation of why it is cheaper to purchase a new printer then replace all the ink cartridges in your current one.