You can do what you suggest, but it has drawbacks. The flat of cuttings will not have roots in two weeks, and will not develop strong root systems for several more weeks. That means that the plants will not be established and growing strongly for a couple of months after you took the cuttings. The first flat will obviously be set back by removing the cuttings. In future years you will obviously be able to take as many cuttings as you want from the plants before the frost gets them.

If cost is a serious issue, then perhaps you should try seed. Typically a pack of seed costs the same as a flat of 6-12 plants. Including soil, you can grow an almost unlimited number of plants for the cost of a few flats of bedding plants.

Ok, I very strongly disagree with "shrubs_n_bulbs". I seldom disagree with him, so he is probably right and I am wrong, but here goes my advice/experience:
impatiens root very very easily. You can root cuttings in seed starting mixture or simply in water.

If you root impatiens, then you can get flowers in 4-5 weeks. 2 weeks to get roots is right, but 4-5 weeks until flowering. at that time, the impatiens have all the roots they need to establish themselves.

if you make the cuttings indoors in 70-78 deg F temperatures in subdued light (e.g. behind a sheer curtain, or under the light of a fluorescent light bulb) the impatiens will root in 7-10 days. You can either fill a pot with a well draining seed starting mixture (available at garden centers) or use potting soil. You can fill a 6" pots and put several cuttings around the edge of the pot spaced a few inches apart. each pot should hold 4-5 cuttings each.

Enclose the cuttings in a plastic bag and if the seed starting mixture/potting soil was very moist then there should not be a need to water again until the impatiens root in 7-10 days.

You can also fill a pan with water and put cuttings around the edge so they do not slide into the water (make sure no leaves in the water.. only the stem). Change water every few days but keep everything at 70-75 deg F.

Once roots appear and they are 1.5-2" long, repot in seperate pots. it may be helpful then to place the pots in enclosed plastic bags for a few days to let the cutting recover. they are usually a bit under the weather (no pun intended) after being repotted from growing in only water.

We always makes tons of cuttings each year this way. I can only start so many seeds indoors, but once one flat of impatiens go outdoors, I make more cuttings for family and friends from these.

Generally, impatiens grow and flower 4-5 weeks after you make the cuttings. We have never had a problem with undevelopped root systems.. not for a fast grower like impatiens..

Extremely easy to root.

I dropped one of my plants and two sections broke off. I put the larger one (with plenty of leaves and flowers) in a jar of water and put it on top of my water heater. The smaller section I removed all but two leaves, dipped in rooting hormone and planted in a small clay pot of soil. At first, the leaves wilted, so I put the entire thing in a plastic bag. The leaves popped back up.

That was four weeks ago. The one in water has five-inch roots and the potted one is thriving with new leaves. I'll plant them outside this week. I plan to propagate more this way. If I can do it, you can do.