This would put us back to the pre-dark world where the government had access to all digital records that were not maintained by people with super-human op-sec practices.
Further, even if you do have amazing op-sec, they could still attempt to prove beyond a reasonable doubt that the data is an encrypted message through non technical means. For example, if they can show that you accessed the alleged data the week before.
Even if there are some cases where, through technical means, one prevents the government access to the data, the number of such cases is still smaller than the number of cases that would be prevented by default encryption.
You haven't addressed the civil liberties angle though: should the government be able to make this software illegal? If not, crypto people can design such a system.
Existing warrant processes already get you to the system that you describe, so I don't know why the House would talk about it as some future innovation and change needed, if their goal wasn't to make the system I describe less permitted.
Existing warrant processes do not get you to the system I describe, because it is not settled law yet whether or not the government can legally compel you to surrender your key, even in cases where there is no dispute that there is a key.
The fact that there exists hypothetical software that would allow someone to plausibly dispute the existence of the key is not relevant to the above question.