A couple of Tempo Tales.
1. I played in a 8-piece band a number of years ago that had some timing issues. Some in the band just struggled with triplets and anything off-beat. If it was 1/4, 1/8, or 1/16 note they were fine, but add a dot and we're in trouble. As a simple test I had them all clap a 1/4 note beat to a metronome. They all were fine. Then I had them clap on the off-beat. Fully half of the group could not do it. And it was the same subset who struggled with off-beat playing. All of the members of this group were mature people with years of musical training. Yet, the fundament ability to sense the off-beat was not part of their abilities.
2. In church we have several drummers. Our most experienced player has been playing since the mid 1970's. He is known for speeding up everything. He'll click a temp at 100 bpm then begin playing at 108 and then speed up to 112. During the bridge when the drums are out, the leader will bring the tempo back down to maybe 105. Drummer comes in and we're up to 115. He knows about his tendency to speed up but apparently either can't fix it or doesn't think it is an issue and hasn't tried to fix it. Nice guy, but not the nicest to play with.
About #1: This is a great observation and I think the suggestion here about tricking out the metronome at 1/4 time and working on consistency between "ones" is a very good one.
I have talked with him about staying on "the one," and in particular explaining that this can mean that you have to listen to everything in between each one and reorient to one in every measure based on the feedback you're getting from the ensemble. He's gotten much better at paying attention to the music as a whole, but this is a work in progress.
About #2: Internalizing tempo is a big problem for many people. He also used to do this same thing but I have been advising him to MOVE as he counts off and to MOVE his body when he hears someone else count off. This helps.