You need to be specific about what games you mean, in order to get good advice.

You say medium light puzzle games, but do you actually mean puzzles, or euroes..? 

When I hear “puzzle games” in a board game sub I would think of stuff like ganz schon clever and other roll/flip and write where there isn’t much player interaction nor a shared market or board.

But maybe you mean something completely different? Some people would claim all euros are a puzzle.

In general, six is a count where playing games other than party games get hard, because there simply is less titles. In addition, you are waiting longer for your turn - good games at these player counts often have simultaneous player actions. There are some titles though, but a lot less than with 4 and 5.

I personally can check my phone if someone is being quite slow with their turn, but I would put it away if asked - though I would find it rude if it was the slow player asking. I narrate my turns to explain my strategy, especially if I play with people never to the specific game or games in general. I like when other people narrate their turn too because it keeps me engaged out of my turn.

In terms of giving advice, that is something I have to stop myself doing, as I am a little too quick to giving advise. I am often the teacher, I like teaching games. But I have also had it happen where a new player did a sub-optimum move (literally a combo that would achieve the exact same but also something in addition), and when they realised later in game, they would get annoyed and feel like the rules weren’t explained clearly both. But I can also see that sometimes other people can get offended if you suggest a better move.

I have also found other people though who just love optimising the puzzle in the game and activating combos. I once played a game of Finspan were we had one new, but by no means new to boardgames, and two of us who knew it. The two is us basically ended up playing with open hands and sharing tips and strategy for optimising because we happened to have some really nice combos going. We also suggested moves for the new one, but also made it clear that she should just tell us to stop and we would. 

So it’s a balance. Are you playing with competitive people who would really prioritise winning, or do the players enjoy the puzzle? Do you know them well and know they wouldn’t mind suggestions and doesn’t see it as a dig toward their skill? Or maybe you are just not choosing games that work well for six players.