I wonder how well I would be able to do those things, though, because I don't have nor utilise a Mic.
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That does make it harder, certainly. Not impossible - very clear instructions ahead of time (assuming you can hear the team chat audio) can count for a lot. In the case of no audio, I think probably the Crota's End raid, on Normal Mode, is easier to handle than the Vault of Glass without comms. Crota's End is much more straightforward than Vault of Glass - you run, you kill enemies, you run some more. The final fight requires a small amount of coordination but it's doable with signals (e.g., "So-and-so Points" or "So-and-so waves") if need be.