So, my question is: why isn't there a standard solution to this problem, designed at least 20 years ago, and implemented since then by all RAID systems worth the name?
The phrase you are looking for is "import foreign RAID configuration"
Not all RAID systems are the same in the way they work, many store metadata on the disks so the controller an rebuild the array configuration if it is reset or replaced (with the same model).. and there is a massive amount of variation between controllers and their supported features.
Imagine I have a RAID5 array with a 1MB stripe size across 12 disks on an Adaptec controller, and the controller fails, so I replace it with an Intel controller hoping to get my data.
Well it turns out the Intel controller does not support 12 disks in an array, only 8, and it has a max stripe size of only 256KB. Of course it is not going to work. Even if it did support those, the controller metadata on the disks is not even close in format to allow them to be read. And even if it did, does it use the same parity algorithm? The same stripe alignment?
As far as I know:
There is no standard on where to store array/controller metadata on array disks
There is no standard on what format to store array/controller metadata on array disks
There is no standard to even store array/controller metadata on array disks
There are many different ways to do RAID, the internal method may be proprietary
And why should there be a standard solution? The solution is simple, buy another controller of the same brand that supports import from the old controller model. There is not much incentive for competitors to reject their own methods or neuter their feature sets in the name of compatibility, when in reality the lack of compatibility is not a big problem.
Now, you had several examples that point out on-board RAID, there are two types here, consumer and professional. Consumer on-board has no interest in portability, unless it is the same chipset vendor, I have gotten Intel RAID5 from one motherboard to another, it reads the metadata from the disk and rebuilds the configuration. Professional on-board almost always expects that you will just replace the motherboard or server with an identical model, and once again it should just work. Many on-board pro grade controllers can also be purchased in addon card form factor, giving you another option.
As Peregrino69 answered, RAID is not backup, if you are trying to use RAID to protect against something other than disk failures, you are doing it wrong.
RAID is more useful at providing large increases in both logical volume size and performance when using multiple disks together, and doing it in a semi fault-tolerant manner.