Isn't it illogical to give more of insulin for a deficit amount of receptors?
Seems like there is some confusion in the definition of type-2 DM itself. According to the American Diabetes Association:
If you have type 2 diabetes your body does not use insulin properly. This is called insulin resistance. At first, your pancreas makes extra insulin to make up for it. But, over time it isn't able to keep up and can't make enough insulin to keep your blood glucose at normal levels.
So the problem in type-1 DM and type-2 DM is quite same; both because of less insulin present than required, the difference being that in type-2 DM the cells need more insulin than normal. And that apparent deficiency of insulin is overcome by insulin injections. Also, the definition you cite doesn't say receptor deficiency anywhere, it just says decrease in receptor's response to insulin.
So why don't we give people a medicine which just increases the number of receptors or increases the sensitivity of receptors?
This is not quite possible yet. Increasing the number of receptors means increasing the gene expression for that receptor, something that is called 'upregulation'. Now, just to add, the cause of type-2 DM has been regarded to be downregulation of insulin receptors on cell surface by high concentration of insulin. See this article:
The process of downregulation occurs when there are elevated levels of the hormone insulin in the blood. When insulin binds to its receptors on the surface of a cell, the hormone receptor complex undergoes endocytosis and is subsequently attacked by intracellular lysosomal enzymes. The internalization of the insulin molecules provides a pathway for degradation of the hormone as well as for regulation of the number of sites that are available for binding on the cell surface. At high plasma concentrations, the number of surface receptors for insulin is gradually reduced by the accelerated rate of receptor internalization and degradation brought about by increased hormonal binding. The rate of synthesis of new receptors within the endoplasmic reticulum and their insertion in the plasma membrane do not keep pace with their rate of destruction. Over time, this self-induced loss of target cell receptors for insulin reduces the target cell's sensitivity to the elevated hormone concentration.
Now, coming back to the main point, it is not yet possible to inject transcription factors in every cell of body so that more and more receptors can be produced. It is possible in vitro, but not in vivo yet. However, it is possible to inject something else instead of insulin which might have similar effects on the body, something known as insulin agonist. Researchers are already trying to create insulin agonists, but haven't achieved much success. See this paper for example, in which they talk about a peptide S961. The problem here is that most of the molecules synthesized act as both agonist and antagonist based on different concentrations. That is why this field has not seen much success.
EDIT: As mentioned by @Don_S in comments, some drugs reportedly do increase insulin sensitivity and number of receptors in cells. Though they have indirect mechanisms for this, unlike insulin itself, but this works too (at least in vitro). One of them, thiazolidinediones, activates a nuclear receptor PPAR $\gamma$, which in turn increases transcription of proteins involved in glucose and lipid metabolism and energy balance. Another drug, pioglitazone, works even in vivo and has been suggested to work by activating receptor kinases and thus increase insulin sensitivity.
Thus, in future, it is very likely that type-2 DM patients would get prescriptions of such drugs instead of insulin injections, but not until there are more such drugs available and at an affordable cost.
Type 2 — American Diabetes Association
Downregulation and upregulation — Wikipedia
Knudsen L, Hansen BF, Jensen P, Pedersen TÅ, Vestergaard K, Schäffer L, et al. (2012) Agonism and Antagonism at the Insulin Receptor. PLoS ONE 7(12): e51972. doi:10.1371/journal.pone.0051972
Hauner H. The mode of action of thiazolidinediones. Diabetes Metab. Res. Rev. 2002;18(Suppl. 2):S10–S15. doi: 10.1002/dmrr.249.
Kobayashi, M, Iwanishi, M, Egawa, K et al, Pioglitazone increases insulin sensitivity by activating kinase of insulin receptors. Diabetes. 1992;41:476–483.