Who would really be the first fighting force to be sent in?

Mission dictates.

Aside from the bomb-droppers as you took note of, for ground forces, if the Marines are the best weapon to wield in a given conflict in a given area, than they'll be the first boots on the ground. If the Army is better suited to first step into the breach, that's who goes. Rah-rah slogans aside, there's no set rule on who must be first to fight. Use the best assets available.

It is scenario-dependent, but if we're talking a MTW with a near-peer, first across the line will be small numbers of SOF to set the scene for USAF B-2, F-22 & F-35 to open some lanes.

Why is 3rd MARDIV conspicuously absent from the graphic? We’ve been continuously forward deployed since the 50’s

Sir read COL Williams- forces used will depend on METT- therefore until mission designation it would be up in the air as who goes 1st

Depends on the mission, delivery and extraction methods available and enemy situation. Looking back through history there is no "one size fits all" answer.

Would it have to depend on the mission and location?

Air Force may be the first to fight but not necessarily the first ones on the ground (except maybe Combat Controllers to set up a forward Air base).

Depends what you mean by fighting force. In any situation...ANY situation that presents itself today, you can be guaranteed that the first US force to be in a hostile country is from the Special Forces Community. SEAL, SF, Delta, other units to be left unnamed. Assuming that they are taking action and not just flying over the place. Now, if you count just flying over the place, then you likely will have some form of Air Force reconnaissance going in before the insertion, so then they would be the first. Unless you count Space Command...because they have been looking at said hostile country for the last 24 months building intelligence. If you get back to effects on the ground though...then it's in the SOCOM community.

In the world of combined arms combat and with the each of the services having many capabilities it is hard to say who would be first in. It would depend greatly on the mission and the plan.

It is safe to say we may have special operators in earlier than anyone else for reconnaissance and other covert activities. Certainly, depending on the target, Naval gunfire would come in early as would air assets from carriers or from more distance bases for heavier bombers.

When we decide to put troops on the ground, we have lots of options depending on the mission. MEU's are all over the globe and can have Marines on the ground in a few hours. For larger operations Ranger battalions and Airborne assault might come into play.

SOF are generally there long before any of the traditional military units, IMO. So... probably Army (Rangers or Green Berets).

During the Cold War..........We always sent in this guy first (Mr Pither) from the USAF.......to see what would happen to him first. Then sent in the rest.
https://www.youtube.com/watch?v=zYV-qYeWPkk