The gap in meaning, I might guess, is due to the absence of metaphor.
Metaphor is the concept of understanding one thing in terms of another - or constructing an analogy between two things, ideas or actions.
Picking the right team building game for example, is not just a question of what the facilitator likes or feels competent delivering; it is selecting a game that provides a platform to explore some of the key issues that the team has, and creating a metaphor in a game that lets team members identify them, work through them, test options, discuss them based on the behaviour in the game, and then draw lessons or ideas that can be useful in their daily work.
Even a quick activity, like an icebreaker or introductions, can be linked to a useful metaphor too. For example, I recently used Thiagi's Hello game to both have people collect information about their experience, expectations, etc., which was good insight in itself, and then in the debriefing asked the group to think about how the exercise might be a metaphor for their work. This game features a number of small groups concurrently collecting information from the whole group in very short segments for planning, collecting, analysing, and reporting of around 3 minutes each! This particular group had some issues that team members wanted to explore about dealing with time pressure, with cooperation and information sharing, and this game was perfect for both introductions and to begin to lightly focus and reflect on these things, even in the first 15 minutes of the day.
Think about where you can find or create metaphor in facilitation and training work. Any extra design element, no matter how small, that makes the link between the activity or game and the work that people are doing (or hoping to do better) can deepen the connection and the learning. And of course, it is important to bring attention to the metaphor, through debriefing, questioning, noticing. Your role as a facilitator is to help people see and make those connections. When done with skill, this helps makes both the meaning of the activity as well as your choice in introducing it much more obvious to participants. Finally, it optimises the time and refreshingly gives people permission to play again ("serious play" of course).
Some groups might need some extra work to help regain credibility for experiential learning. By strengthening the metaphor and meaning of games and activities, you are both investing in a group's future success learning together through interactive techniques, and also hopefully softening resistance, making your life easier on the day and afterwards.