What adapter do I *need* for this - or does it exist?

['Pants]

You all know those saddle-kits with needle-valves for icemakers, right? Well, I'm fed up with the low life-expectancy of the 1/4" poly compression outlet setup, so to convert to push-lock, I went looking online for what I believe would be called out as a
1/4" female compression x
1/4" push-lock
adapter. I want to do away with the female compression-nut with the long inner sleeve and maybe-one-time-use ferrule (etc) in favor of a pushlock type tube connection at the outlet of that valve.

In way too many searches already, the closest description I've found to that is called out as "Tube to Female Flare". (see https://www.amazon.com/Parker-PP4FF...BSEANSPP2SF&psc=1&refRID=6CKN33CT6BSEANSPP2SF )
The term "Flare" makes me think this is wrong. The "Tube" part would be the push-lock side.

In fact, I'm not convinced that I've found ANY fittings, other than a cap, that would thread on to a male compression fitting and provide a seal. What am I missing?

 

[Jadnashua]

First off, if you want a reliable install, ditch the saddle valve entirely. You could put a T in the line and put a decent valve and run that to whatever type of tubing you need (which would depend on the tubing you need). Many of those are compression fittings. Once you have a decent valve in there, you can use whatever connection to it that you want.