Note: This answer was posted 4 months before the OP clarified what it really wanted to ask (see comments below the answer).
I will accept @Ubiquitous view of the question, which in summary is:
Why not having a publicly owned monopoly in the banking sector? Instead of a, however regulated, private banking sector?
It would be naive to counter "then why not nationalize all production activities?". This is not a "socialism or capitalism?" socio-philosophico-ideological issue. Strictly from the point of view of how the Economics discipline analyzes things, we must recognize the critical and special role the banking system plays in modern economies (because it intermediates savings and investment, consumption smoothing and intertemporal allocation of resources), as well as its peculiarities.
And whenever the words "critical","special", "peculiar", arise in economic analysis, the merits of a private-market structure against a publicly controlled one must be carefully analyzed and assessed.
First let's define what is the main business-line of a bank, in the most basic, abstract sense: sublessor of perfectly liquid capital. It borrows capital from the savers and lends it to the debtors -for a fee, which is the interest, and which depends on the length of the lease. And as with any lease, the lessee must also return what has been lent.
Let's move now to some of the main reasons that have made societies produced a good/service through a state monopoly (obviously I won't go through all the taxonomy and intricacies of the various categories of public goods):
Non-excludability pertains to public goods like national safety: we cannot exclude those citizens that don't pay for it. This creates disincentives to pay for it. But the society as a whole needs it. So everybody pays whether they want it or not (through taxes that are backed by the monopoly of violence that the state holds), and the service is offered to all by a state monopoly. Does non-excludability holds for what the banking sector offers? No, because the bank gets to decide who gets the service or not. And the "service" is then at the control of the debtor. So this cannot be an argument in favor of a public monopoly in the banking sector.
Decreasing Average Costs with the scale of production, is what traditionally defines a "natural monopoly": it is in the nature of the production process to have decreasing average costs as quantity produced increases, and so the state steps in so that the benefits to the monopolist accrue to all society through the state. Is the business of a bank characterized by "automatically" decreasing average costs as credit expands?
What are the "production costs" here? Various administrative costs certainly (branches, overheads etc), which are unlikely to exhibit a downward trend through out -rather they are very well likely to have the familiar U-shape, falling for initial small levels of credit volume, then perhaps a stage of constant returns to scale, then decreasing returns to scale (and so increasing average costs). A distinct and peculiar cost here, is outright loss of the capital. Surely, when one lets an apartment, it may suffer various damages from the use of it by the tenant -but it won't disappear with him. These costs will almost certainly increase with the "scale of production" (credit expansion) as the bank expands to less trustworthy debtors. So no "decreasing average costs here" - the banking sector is not a natural monopoly.
A case of negative externalities can be made, that moreover cannot be reversed: the choices a bank makes as to where it will lend its capital, and for what purposes, may not satisfy the needs of the economy as a whole: the economy may need transportation infrastructure -but the banks may lend capital for building office space, which are less needed. But once capital is sunk into buildings, it cannot be transformed into roads (certainly not directly, while indirectly, it won't be timely and it will have dead burden costs). But externalities are regulated in other ways, not by imposing a state monopoly.
What about the various imperfections that inadvertently characterize all actual markets, banking sector included? Incomplete and asymmetric information, rigidities, straightforward inefficiency, etc? Here, the matter becomes quantitative, exactly in view of the criticality of the banking sector: a given level of imperfection and inefficiency would matter less for society if observed in another market, compared to the banking sector. So we have to measure it -and then compare this with the level of inefficiency and imperfection that we would observe if we had a public monopoly instead.
This is what I had to offer as food for thought. I close with some responses to the OP's specific projections:
(In case of a state monopoly, the businesses would have)
A lower interest rate
Why? The central bank would lend public money. To private businesses. It would also have to bear administrative costs. Also, it would be accountable for public money. This would introduce conservatism into the system, to avoid accusations for favoritism for example. Either higher interest rates, or much more stringent criteria to grant a loan. Would that be desirable, as there will be no-one else (officially) lending capital in the economy?) What about the possible increase in loan-shark activity?
Control on their interest rate
I do not understand what that means.
Theoretically no limit
No way. Either the OP has in mind that the central bank will have vastly larger amounts of capital to lend (it will, but will it lend it "easily"? -see previous remark), or he has in mind creation of capital out of thin air based on the "state's authority", which of course would have solved all humanity's materialistic issues, if it could happen.
