This is a common issue when sailing in restricted waters. For instance if Turquoise could give room because of the river bank, or a line of moored boats.
Red does not have room so must keep clear under RRS 11.
If Red gybes on to starboard then RRS 15 applies and she must give all 4 boats room to keep clear, including  space to comply with their obligations under Part 2 and 31

This doesn't necessarily look to me like they're unable to comply.  Obviously depends on boats, windspeed, and other factors.  

My first question would be if they're unable to manuever to allow mark room, what keeps them from slowing down? If it's a 49er or I-14, clearly not possible.  If its an IOD/J-24/Laser, why didn't you douse/overtrim/etc.? And as they're drawn extremely close to each other, one would assume this is in very light winds, which would further allow slowing.

Second thought would be that while it may absolve yellow, would it absolve the other three who also had an obligation to give room if possible?

Red should never expect that by getting room at 3.5 boat-lengths that she will suddenly have THE POWER (except maybe in teams racing or match racing). That applies doubly when there are multiple boats. Red should have been planning to have no room at position 1, when she was clearly astern and at the whim of Yellow's angle, as well as the maneuverability of three additional boats who cannot see her.

Flick onto starboard at 1 and sail a hot angle to make the dominoes fall before the zone, and then flick back to port into that gap?

There’s the plan, and then there’s what the wind allows, and the two sets may intersect if you’re lucky…

Agreed, which is why you have to do it early at 1. It’s your one chance of making the others make it their problem rather than all your problem

You’ll even then likely not get all of them, and you’ll be in trouble if you push for all of them, but by going early and taking it handy on a broad reach, you’ve got a sporting chance the outside two will be driven wide enough you’ll have a decent lane to slot into. 

Well that's even more clear that they could slow to give mark room.  

Referring to the first diagram.

Red is required to keep clear of all the other boats to leeward of her on the same tack (RRS 11).

Red is not entitled to mark-room (RRS 18.2(d).

As diagrammed, Red is keeping clear and breaks no rule.

If Red had tried to get between Yellow and the mark, she would break RRS 11 and not be exonerated because she was NOT sailing within any mark-room to which she was entitled.

From @4, if Red does not turn back and leave the mark to port, she will [eventually] break RRS 28.  For a breach of RRS 28, there is no on-water penalty and Red's only penalty is to Retire or be scored NSC by the race committee.  Red is not 'at fault' until she crosses the finishing line without rounding the mark on the specified side.

Red needs to tack around and round the mark on the specified side.