This is just not true. Drug companies will make slight modifications to existing drugs and can get new patents on the new formulations and for new indications. If it works, a drug company can make money off of it even if it's been off-patent for years. See https://en.wikipedia.org/wiki/Lenalidomide vs thalidomide for an example.
For every therapeutic target there are many chemical/biological solutions that can work. If an off-patent compound shows some effect in a pathway, a patentable variant can usually be made.
How can you say this is not true then give all example that require a patent? Yes you can mine drugs that are out of patent and find new uses for it that can be patented. Do you know of one new drug that has come to market in recent years without patent protection?
The difference between (a) an existing, off-patent molecule and a (b) patentable variant of that molecule is somewhat trivial compared to all of the other costs of bringing a drug to market.
But to address your specific question, generics are drugs that come onto the market all of the time without patent protection.
There's plenty of drugs that are made by multiple companies. It might not make sense for a very specific drug which sees low use, but I think it's viable to produce a drug that might be beneficial to virtually anyone even if you don't have exclusivity.
Yes once a drug goes off patent and becomes generic then lots of drug companies will pick it up and start making it, but no new drugs are developed these days without patent protection.
Of course if you wanted to you could make this treatment yourself - it is not that complex.
The issue is who is going to pay for the extensive testing required to get from lab to hospital. That takes years, and lot of organizational skill, and a lot of money. If a drug has gone off-patent, all that is already done, so many companies will be glad to manufacture it. But for new treatments, somebody has to push it through clinical trials, with the (quite substantial) risk that something which looked good in the lab fails to work in larger studies, or has unforeseen side effects.
For every therapeutic target there are many chemical/biological solutions that can work. If an off-patent compound shows some effect in a pathway, a patentable variant can usually be made.