Yes Avidemux handles all the formats. When prompted to save the file after editing, one has the option which format to save it in. If in the input format, the save is instantaneous. If in another format, it takes the time required for transformation.
I prefer to do just the editing in Avidemux and then any transformation (like TS to MP4) in Handbrake (which gives me more options and control).
Now if the cut is on a non-key frame, the transition between segments is not that smooth. So at times I’m willing to sacrifice one or two non-key frames to have an absolutely smooth transition between cuts.
Unfortunately the transition is so smooth that my wife has taken to scene removal (asking me to remove those scenes that upset her in a film). One wouldn’t know there was a cut unless one knew the movie very well. So in one film, she asked me to cut out a scream and all one sees is a person ready to scream and then smoothly another person just simply walking away from the screamer - no scream ever heard, just a transition from one person to another LOL.
I also like the Avidemux function of substituting one audio track for another. I did many Christmas (fireplace) videos replacing the original audio with a selection of our own carols.