No, not possible. The OP stated voltage reached 13.67V and stayed there. The charger NEVER got into bulk. And... 13.67 is not a specified float voltage. He also tried AGM mode with no change in output voltage. The charger is defective. I would not trust it.
It's highly improbable you'll see anything abnormal with an oscilloscope. Another voltmeter could confirm initial readings.
One wouldn't know unless they had watched the entire charge cycle with amps and volts monitoring. Bulk is CC, so it would hold the amps constant, volts may come up after some time to specified number, and when it arrives at xx volts, it signals switch to float CV (holds voltage), and lets the amps balance in until no more current flows at that voltage.
So without watching an entire charge cycle while monitoring, then it can only be implied that it didn't do its job if we're only going off of a couple sample-points given by the OP. A scope reading both amps and volts could capture a larger sample of the charging cycle behavior in order to arrive to a more specific conclusion.
I'm not saying you're not correct, but we don't know if that is what is occurring here without collecting more data.
If he really thought the charger was not right, then could exchange it for a replacement of same model, and observe the behavior of that one. I agree that if it is only floating the whole time, it could take a long time to charge a battery (if it isn't able to do bulk CC for the first majority of the charge cycle), but we don't have enough data to know if that is what is happening here.
If he has an amp-clamp he could measure current as well as voltage and watch it on charging, to help decide if bulk is working right.