Apologies for the long response, and thanks for the comment.
I think that would be a great solution.
While I've not yet gone through the steam process, I'm aware of what you describe.
The 2 big problems for itch I can see in implementing this style of solution is that to register a signed piece of software is quite expensive generally, and that itch would need to be able to introduce a system for reviewing games to ensure that, as much as possible, only "safe" games are then given the right to use itch's certification.
Without considering these points, there would be a lot of risk to itch if people misuse any offering.
At the moment, itch is essentially a free marketplace for both learning and independent developers. For itch to be able to implement a solution to this might mean they would need to change their approach to be able to fund such a solution.
My personal opinion is that there needs to be a more affordable approach to certifying software, whether it is through itch or the companies that manage the registers of signed software, because at the moment genuine hobby developers are priced out of the market and have to rely on trust at a personal level for others to use the software we create.
For example, I'm a UK based hobby developer with no savings and a low income job, the cost of uploading a game to steam, while not prohibitive, is enough that I would only be willing to do it if I make a game that I think will really have potential. The cost of signing the software myself however is too expensive for me to even consider it as a viable option right now. This leaves me stuck with a situation of needing people to be able to play and pay for my games for me to be able to invest in signing the software as appropriate, but people will generally not pay for software that is not signed as the host system will view it as a security risk, regardless of the actual behaviour of the software, preventing me from getting people to play my games as downloaded software as there is no trust or reputation.