So how would you want the game to proceed in a basketball-like manner once a player dives and secures the ball? Is there a reasonable method by which that player can get off the floor with the ball without taking 2 steps? (I figure it's possible only if the player's feet were not touching the floor at or subsequent to the time s/he got possession.) Or is that player's only feasible option to try to pass the ball from that position?

The only game I can think of with a similar question is Rugby Union, in which a player is allowed to dive for the ball but is not allowed to remain on the ground with it or in its vicinity. The referee looks for whether the initial dive was a bona fide attempt to get the ball, whether the player stays on the ground with or near the ball to gain an advantage once there, and whether opposing players are preventing that player from getting up or crawling away from the ball. Unlike basketball, there's no "steps" provision, so the player on the ground with the ball may either pass it, leave it, or get up with it -- but there is an analogous "traveling" provision in that the player is not allowed to crawl with the ball.

Could basketball be construed similarly, with being on the floor with the ball as an illegal position only if the player remains there without trying to get rid of the ball? And being on the floor without the ball being illegal only if the player remains there to take advantage of that position?

He can start a dribble and get up.

Options ...
 

Also start a dribble (if not having already dribbled), try for goal, or request a timeout.

If flat on back, may sit up without violating.

In forty-plus years, I've only had single player attempt a try for goal from the floor.

Unfortunately, he missed.

180 degrees
 

Or as was discussed, in one of my previous posts, the player with control of the ball lying on the ground may roll no more than 180 degrees :D

Mr. Fiore, My High School Geometry Teacher ...
 

... taught me well.

Which is exactly why I carry a protractor in my pocket along with a spare whistle.

https://snoopn4pnuts.com/cdn/shop/pr...g?v=1658606997

I was 1 for 1 in such attempt in Columbia's gym. A fellow student said I should do ads for NY Telephone.

Meanwhile, didn't we have someone here point out that you can't start a dribble without a foot on the floor? But I guess nothing says you have to have weight on that foot; it's not like wrestling rules with supporting points.

Could you roll off the ball to set a pick, like how I set a ruck (illegally, because I never touched the ball in my dive near it) in rugby?

On The Floor ...
 

Wait? What?

B1 jumps and catches a rebound after a try and while B1 is still airborne B1 starts a dribble by pushing the ball to the floor.

Legal.

Well, I remember there was some nonobvious condition to starting a dribble that was brought up by someone here.

Starting A Dribble ...
 

2000-2001 NFHS Basketball Interpretations Supplement #1 Situation 1: A1 is an airborne shooter preparing to release the ball on a shot attempt. Instead of releasing the ball on the try, A1 fumbles the ball (while still in the air) and drops it. A1 then returns to the floor and secures possession of the ball. Ruling: Traveling violation. While airborne the ball must be released for a try or pass.

Also (from Misunderstood Rules list):

If a defender puts a hand(s) on a ball controlled by a shooter (or passer) who is airborne, such that the airborne player is unable to release the ball, it is a held ball. If, in another similar situation, the airborne player loses control of the ball because of the touch, then this is simply a blocked shot. It is not a violation for that player to start a dribble at that point. If, in a similar situation, the defender simply touches the ball, and the airborne player chooses not to release the ball, and returns to the floor holding the ball, it’s a traveling violation.

When an airborne player tries for goal (or pass), sees that the try (or pass) will be blocked, purposely drops the ball, and touches the ball after it hits the floor, that player has traveled by starting a dribble with the pivot foot off the floor. If an airborne player is preparing to release the ball on a shot attempt (or pass), and instead of releasing the ball, fumbles the ball (while still in the air), drops the ball, and then returns to the floor and secures possession of the ball, it’s a traveling violation.

So if you gain control of the ball by diving for it and have landed with neither foot on the floor, don't you have to put at least one foot down (i.e. have some part of it touch the floor) to start a dribble?

OK, then, after doing that you could slam the ball down hard enough that it rises to give you time to stand and catch it.

No.

The pivot foot cannot be LIFTED before starting a dribble. If it was never on the floor, it can't be lifted.