There's nothing wrong with 3 shot groups provided they stay in the same place and you shoot enough of them. If you don't, they will not tell you much that's useful.
The barrel vibration consideration is real an can quite easily be what you are seeing. Some rifles will have issues with tuning a load. I have a 25-06 that does what you describe but throws bigger groups with Barnes 100 grain bullets. Careful testing with that rifle established it's preferred seating depth with those bullets, but no way in hell could I bring that rifle below an inch after trying half a dozen or maybe a couple more powders as long as I tried to bring the velocity in at near max pressure. It always tended to 2 and 1 groups and they wandered some. I bought some factory ammo for super cheap to get the brass and bullets. I decided to shoot some to see what it did. It shot very nice groups that didn't wander, but at about 250 FPS below what the rifle is capable of. Measuring showed the exact same seating depth I had found the rifle liked. Testing at that velocity with several powders proved any of them could produce groups down near a starting charge.
Pretty reasonable evidence that barrel vibration was the issue. I tried a rubber donut on the barrel to see if maybe I could compensate and get my 250 FPS back, but no luck.
I would look at seating depth as an issue first if you are confident of a good load. Load development can get expensive if you have problems, so, just look at your fliers as part of the group and assume those fliers are part of the group and that a 5 or 10 shot group would maybe have half of it as "fliers". If there' any evidence of the group wandering, that has to be considered as part of the group as well. If you are certain of the seating depth your rifle likes then work with the charge and go carefully from starting load up to max. If that doesn't solve it, try a different powder. Sometimes just moving that pressure peak just a little earlier or later will look like a miracle just happened. I have never found a way to illuminate my guesses about whether a slightly faster or slower powder will solve my problem(s) Similarly, with monos which I have found can be very touchy about seating depth in some rifles. Cheapest for me has worked out to be if they won't shoot at the starting depth I just move up close to the lands and then work back in .010 increments. Some rifles are easy. Some rifles are hard. I have learned that it's not my choice which rifle I am working with. About 2-4 per cent of the rifles I have solved are "difficult" if you want a given rifle to shoot a given bullet. Sometimes it's easier to change bullets.
