There are ways to make perfectly secure (from a mathematical standpoint) backdoors. Theoretically, there is no reason that every encryption couldn't have a second key known to the government that doesn't weaken the encryption by any significant standard.
The problem is about trusting the government, which can mean (at least) two things. Either you don't trust them to use the key properly. Which is a valid concern. Or you don't trust them to keep their official backdoor key secure. Which is also valid, secure data is stolen all the time and we can't even be sure that they'd notice or admit if it did get stolen.
Again, the technical challenge here is that in the real world keys can be stolen. The "system" can be secure from non-government actors even if the system comes installed with a backdoor that the government has access to. The weakness isn't in the theoretical "system", it is in the fact that now instead of one point of weakness (you yourself protecting your own key) there are two real-world points of weakness (your key and the government's key). The "system" is still impregnable to the same types of attacks, but now rather than tricking you out of your key someone might be able to trick the government out of theirs too.
Now hypothetically, what if you divide the "government key" in half. Give half of that key to the vendor controlling the encryption and half to the legal system. As long as you don't restrict the company from speaking out about its cooperation, that would seeming prevent abuse as well as minimize damage if one key leaked.
The problem is about trusting the government, which can mean (at least) two things. Either you don't trust them to use the key properly. Which is a valid concern. Or you don't trust them to keep their official backdoor key secure. Which is also valid, secure data is stolen all the time and we can't even be sure that they'd notice or admit if it did get stolen.
Again, the technical challenge here is that in the real world keys can be stolen. The "system" can be secure from non-government actors even if the system comes installed with a backdoor that the government has access to. The weakness isn't in the theoretical "system", it is in the fact that now instead of one point of weakness (you yourself protecting your own key) there are two real-world points of weakness (your key and the government's key). The "system" is still impregnable to the same types of attacks, but now rather than tricking you out of your key someone might be able to trick the government out of theirs too.