I agree with Sherri and Craig as well.
I've always provided an appendix look up table between the manual and different standards that it covers - especially true for QMS manuals that have had to cover off ISO, CCC (China), UL (USA) and other standards and keep multiple different auditors happy. They just had to look up the cross reference to get to the correct chapter.
Don't go fixing it if it ain't broke.
There's nothing in the standard saying how the manual has to be laid out.

Paper folders with pieces of paper that have to be kept up to date across multiple sites

How about a "days since last safety improvement" sign? It has been zero days since there was a positive safety suggestion. It has been three days since the last near miss was reported. It has been a week since the last group wellness activity. It has been three weeks since the last H&S committee meeting. Focus on the lead indicators.