Some medical doctors and medical borgs already go out of their way to do paramedic job.
I feel like paramedic is always a little bit too close for comfort to crossing validhunting rule. Usually people drag other crewmembers to medbay but when you see someone's sensors goin red in maint as paramedic you can either go in and most likely interact with antag or inform security and most likely give antag time to turn off victim's body sensors and dispose of the body. Back in the day when I was playing mostly paramedic IPC I was actively goin into maints to retrieve victim's body without engaging in fight with antags (but sometimes just trying to steal the body from the antag if they are still present) and I remember I got warned that it got a little bit out of control and was told to stop BCS it got a little bit harder to kill someone without me starting a big ass search in maints for body.
While we could merge the paramedic with medical doctor and just make "paramedic" an alternative title to medical doctor, I believe there's a good reason why we only have one paramedic.
I believe we would want to avoid situation where 10 people are running around station and maints trying to save other people. Messy for security, messy for antags. Could make it easier for antags to kill doctors but would also make it really hard to kill literally anyone else if 5 people were monitoring suit sensors at all times.
I think that idea with people receiving damage if they are unconscious and dragged on floor is worth at least consideration.
I dunno how you would see this balanced tho. If the damage they receive is low then there's still little to no reason for waiting for paramedic. If you make it a big number then if paramedic is busy or dead then victim is just gonna bleed out.