Could be a couple of things. Silicone caulk, when removed, can leave a residue that makes it difficult for other applications of caulk to adhere to. There are "caulk residue removers" available at box stores, that might help.

"Liquified caulk" is usually a sign of 100% latex caulk that undergoes long-term water exposure. This exposure could be either from standing water on the edge of the tub after showering, or from water coming down the "back side" of the tile, or down the drainage plane, and hitting the caulk.

If the caulk is mushing out just due to direct water spray and standing water on the edge of the tub, then going with a siliconized-latex caulk will give you better water resistance. 100% silicon has the best water resistance of all, but it can be a bit messy to install and tool.

As to not caulking that joint, if the drainage plane (usually 6-mil polyethylene or tar paper) that is behind your tile backer board overlaps the flange on the edge of the tub, you might be okay.

Thanks for the feedback. I wondered if I might have used silicone caulk in the past.

The latex caulk has liquified in a matter of days. I suspect water is somehow getting behind the tile/grout and coming in from behind. Nonetheless, I am suprised it has failed so quickly.

I don't think there is either tar paper or polyethlene behind the hardibacker. It is just the hardibacker that overlaps the flange. I remember the contractor telling me that is how it would be installed and I recently removed the panel to see if water was leaking behind the tile and noticed that was how it was installed.

After doing more research, I'm also concerned about another issue. It is grouted in all the corners and were the tile meets the tub - that looks like like a no-no. Also I read in another post that caulk is not there to waterproof - is that correct?

Correct.