Yes...you can not tell if it is tight. The tiniest of vibration under that board can give you grief. You may be fine, but there is no way of knowing for sure. Good Luck.

Now that you've got a clean slate, instead of redoing with backerboard, try Ditra instead. It costs more, but it's way easier to work with, isn't dusty/messy to cut, and provides a better isolation of movement in the subfloor.

If the manufacturer of the underlayment says to put down thinset, I'm not sure why you would not want follow their recommendation. There is also a screw pattern that needs to be followed as well.

Although I'd be more inclined to think that the tiles popping was more a function of not having enough thinset under the tiles. After you lay a tile pull it up and ensure you 85+% coverage. If you don't have enough coverage you have the wrong trowel or are using it incorrectly. Again, carefully read the instructions on the bag and follow them.

The thinset under backerboard is NOT to stick it down - that is what the screws are for. The thinset is there to fill all of the void space so there is no flex when you step on it. Failure to place the backerboard with thinset under it was just begging for failure - exactly what you saw.

Additionally, your floor needs to be stiff enough for tile. What is under your backerboard? Hint - plywood, plywood, plywood. You want 1 1/4" substrate under tile. Then you need an isolation layer to prevent cracks, tiles popping - this will be either backerboard or an isolation membrane, such as Ditra (much thinner than backerboard). Then tile. Do it correctly and you'll have a solid floor that lasts pretty much forever.