Etherscan ABI – Can ABI Be Uploaded to Etherscan Without Source Code?

We could help better if you included more details such as contract addresses. But in your case it seems like that contract is a Proxy Contract, this is why you don't see the implementation nor the corresponding ABI.

And yes, you can manually add the method you are calling to the ABI and use with web3 libraries. Or, you can also use encodeFunctionSignature

In theory, you could create your own code that creates the ABI of a contract given the source code of that contract, reading the function's name and parameters and producing a javascript/JSON output, but you would be reinventing the wheel since you would be implementing something similar to a compiler.

Web3js and Ethersjs don't do this.

You could put the contract source code in Remix and compile it and get the source code. But I guess this is not what you want since it will use solc.

You could create a tool in, like, nodejs to programmatically handle the abi generation given some contract code base, using 'solc' under the hood, but adding more functionality, like receiving source code of contracts dynamically and compiling it, taking the abi and returning it, etc. Like a backend, like Etherscan and Remix do.

Check out this project: https://github.com/smartcontractkit/solc-api