Five teams sounds like a lot; building and sustaining an effective trust-based coaching relationship with more than 10-15 people is a significant challenge.

In your context I’d be looking to grow the leadership skills within the teams so that you can adopt more of a coaching stance; this seems to align with management’s expectations.

What shape that tales is context dependent. Things that have worked for me include

– sending everyone on a two-day “team member to team leader” course, across 2-3 cadres in a department of 50 people; you get about a 20% hit-rate from courses on average, but 10 new people showing leadership in the teams is a big help

– run leadership/agile workshops on an open basis; I do these on Friday afternoons. Work on core areas like facilitation, problem solving, communication, conflict resolution, negotiation, as well as agile and lean practices.

– spin up roll-centric and subject-centric communities of practice, who can take the led in developing taking agile practices forwards some of the high-performance areas covered in books like “Accelerate!”;

– look to bring the teams up to speed in problem solving techniques like “evaporating clouds” or Ishikawa fishbone; use tools like Anthony Coppedge’s retrospective radar to pull together systemic issues for management

– get a mentorship group running outside of your employer; that is to say other Scrum Masters and agile coaches who can support you, and you in turn can support them

In terms of influencing management, perhaps reflect on the skillsets you need to do this, and how you would currently rate those skills. Then look to what professional development may help. Things like leadership training, negotiation skills, courageous conversations and coaching training have all helped me.

Scrum is really designed for one team working 100% on one product, with a single Sprint Goal.
I’m curious why you want to try to apply Scrum in a scenario where this is not the case?

You might be better off working with a Kanban Method approach, perhaps with a swim lane for each product/application. That way you can start to use data to explore how the work is flowing, and help the team to improve.