Discovered a dummy outside air vent

While in my attic looking for a place to run new electrical wires I noticed the air vent coming through the roof down to the upstairs bathroom
was broken. To my knowledge this was the only air vent in this system. I looked around to see if the air vent broke off in some adjoining wall but found no trace of that.
This system ( toilet,tub and wash basin) seems to have been in place for many years and it functions just fine , I have never noticed any kind of slow draining or sewer smells. It is almost a straight run from the upstairs toilet to the waste stack in the basement. When I discussed this with a friend he said it must be venting somewhere! However I see no evidence of this. I cannot actually see much of the plumbing in this bathroom because , to create it , they built a wood step up with all the pipes below this wood structure.

So my question is can a drain/waste system function without an outside vent? I know in theory it needs a vent, but are there cases where it could work without one?

Now last year , when creating a new downstairs bathroom, I added an outside vent thinking it was just an adjunct to the old one. However the system above has functioned just fine for at least 15 years before I bought this house.

From your description, I do not know if you have a vent into the attic that does not go through the roof, or no vent at all up to the attic. But, I cannot see any reason to go to the trouble of installing a "dummy" vent if it served no purpose. A system "can" operate without an 'outside' vent, most of the time, but under certain conditions you will have drainage problems. I am not even sure that you are making the correct diagnosis regarding the presence or lack of a vent.

Sorry I wasn't clear. I have only 1 outside vent on my roof of the second story side . While in the attic I discovered it does not go anywhere. It's broken. I never knew that before because the broken end was buried in attic insulation. Until this discovery I had always assumed it went to the pipes in the upstairs bathroom for venting.
So if I understand your reply the answer is yes, a system can exist without outside venting BUT under certain conditions it could cause problems.
So now for a second question. I did put in an outside vent on my downstairs bathroom. Would having this working air vent prevent these problems?
What is at stake here is if someone said absolutely yes ! This upstairs bathroom must be vented I would have to totally pull apart the upstairs bathroom including the floor to gain access to the pipes. If this system existed and worked for at least 15 years before I added this second vent i was hoping I could avoid this.