Improving the standup
What my question is, how to make this more as a team sync and less as a team rapport?
I don't think there is enough detail in your question to answer that. You've described symptoms (some people feeling like they aren't being heard) but haven't explained the root cause. Are some team members joking around without listening? Are they being disruptive? Are they staring at their phones? Do some team members dominate the timebox?
I'm going to make a guess and say that your standup has become nothing more than each person giving the scrum master (or PO) a status report. That is not the goal of a standup. The team should be telling each other the answers to the three scrum questions. The scrum master and any other stakeholders need to be a fly on the wall rather than an equal participant.
Have you tried literally having the team stand in a circle, with you and any other stakeholders standing outside the circle? Perhaps if every member of the team is facing the rest of the team and with you on the outside, they will feel more like they are talking to their teammates and less like they are reporting to a manager.
If someone feels they aren't being heard, then that means that others aren't listening. Do you know who those people are? Have you talked to them? You should work with those individuals to understand why they aren't listening, and then help them understand why it's important for them to give their team members respect during the standup.
Use the retrospective
The way to solve this isn't to ask for help on the internet. Scrum has retrospectives for a reason. The retrospective exists for the team to solve team problems, and what you've described is a team problem. As scrum master, you should introduce this as a topic at the next retrospective. At the end of the day, this is a problem that the team needs to solve for itself.
I feel that I should remove blockers but ... I don't think that's only SM's job.
You are correct. In my opinion, it is not your job to remove blockers. It is your job to make sure that blockers get removed. Sometimes that means you or the PO needs to step in and actually remove the blocker, but other times it means you need to work with members of the team so that they can resolve the blockers for themselves.