Usually even in countries that fund prescriptions, they will only fund the generics completely. At best, if you insist on a brand name, they will cover the cost of generic and make you pay the difference.
But yes, the doctor IS prescribing "the actual benzo chemical itself". When you talk about generic drugs, you are simply talking about the manufacturer/name of the drug, not the chemical itself.
Basically, when a new drug is invented, the company who invented it gets a patent for a certain number of years giving them the rights to be the ONLY manufacturer of the drug. This is because drug development is expensive so they need a chance to recover their costs. If they weren't given the initial monopoly, there wouldn't be much incentive to develop new drugs. So they'll release the drug with a name created by their marketing people, which is though of as the "brand name". Klonopin/Rivotril is an example of a brand name.
When the patent expires, other companies are then allowed to manufacture the drug. In most developed countries, they do not have to go through the same development costs and studies in order to have their version of the drug approved, but they have to prove to the governing organization that the drug is "bioequivalent", which basically means that their version of the drug is nearly identical in terms of how much of the drug is ultimately absorbed by the body. When they put these "clone" drugs out on the market, they don't usually come out with their own brand name, they just label it as the name of the drug, which some people call the "generic" name. For Klonopin/Rivotril, the "generic name" is clonazepam. The drug is technically the same, the only real difference is that it's being made by a company that does not have the sole manufacturing rights that were granted by the original patent. Many companies that make generic drugs SPECIALIZE in producing generic drugs and don't have much of a research and development arm of their own. So, when generic drugs finally hit the market, there are now a bunch of companies competing to sell the drug instead of just the one "brand name" that has a monopoly and practically gets to set their price. And since there's competition, the prices are driven down.
But a lot of people know the brand name and only trust the brand name, even if the generics are identical. So the brand name can still often get away with charging a bit more. But the vast majority sold after the patent will be generic. In fact, here in Canada anyways (where we pay for our own drugs), if the doctor writes a prescription using the brand name, the pharmacist is obligated to either give you the generic drug instead, or at least inform you that a generic is available and that it's not really any different. There are some exceptions though, such as certain heart meds, where even a TINY difference in bioequivalence which is normally thought to be acceptable, can be dangerous for the patient.
There's also another exception regarding SOME XR/time release pills. Wellbutrin XL is a recent example. The patent on the drug itself is expired, so companies are able to manufacture a generic form of it. But the company also has a patent on the specific XR delivery system used in the pill, so the generic manufacturers have to use different, older systems for the pill, making it technically not identical. Roughly the same amount of the drug ultimately gets absorbed by the body, but sometimes at extremely different rate. Some people that were switch from the brand name to the exact same dosage of generic were becoming nauseous and vomiting because the generic was releasing the drug into the bloodstream MUCH more quickly. This kind of situation is very unusual though and not something that could realistically happen with an IR drug like Klonopin/clonazepam, all the generics should be pretty much identical.