He can’t fail a jump. A 1 is not an automatic failure unless there is a penalty for failure. Therefore there is no chance of him failing 7’ running long jump.
So it’s your roll plus your athletics score is your minimum, any athletics proficient PC will have a 3‘ minimum jump.
As a DM you have to watch how you draw terrain, if you draw a 1’ wide ledge into a 5’ space the the pc can just move from the ledge into the next 5’ space just by taking a step.
I would say falling into a very deep pit would be a penalty for failure.
A jump should be an exciting moment in a game. If it takes more than a moment to adjudicate, it is taking too long. The PHB rules handle it perfectly by giving simple, easy and dramatic - yet quick to adjudicate - rules.
You have some guidelines in the book, the rest is up to you, personally I give more penalties to people with armor, heavy weight, etc and use the xtreme difficulty (25) for world recor distances. In any case my advice is to adjudicate difficulties fast and keep going, don't lose time with that.
Everything I stated in what you quoted is, in fact, taken from the rules. Whether you choose to run your game by those rules is completely up to you. You should not read my statement as an imperative that you must do so, only that this is what the book says. And it takes into account what you're referencing in the Using Ability Scores section in a way that dovetails with the basic conversation of the game and adjudicating actions.
Literally all they have to do is "I use Strength (Athletics) to jump an extra 5 feet" and your job is to assign a difficulty to that roll.
Well...this may seem inconsequential but I think it is very important.
Players don't declare that they are using skills.
Players say what their character is doing and if the DM judges that the outcome of what they are doing is both in doubt and interesting/has consequences then they call for an ability check and may call for a skill to go along with that.
So in this case the player would say "I jump across the chasm" then the DM adjudicates from there.
Isn't it also ignoring the rule which says: You try to jump an unusually long distance or pull off a stunt midjump?
I don't remember if they say it in this edition, but I assume specific overrides general.
But there is nothing in the rules that says that the character making the Strength (Athletics) check to jump an unusually long distance (a thing explicitly allowed by the rules) needs to come up with some convoluted explanation for how they are going to do so. Literally all they have to do is "I use Strength (Athletics) to jump an extra 5 feet" and your job is to assign a difficulty to that roll. Now, a player may say that they are going to find an outcropping in order to give them just a few more inches of height to try and clear the distance, or that they are going to use their spear as a pole vault or whatever it is, but none of those things are required by the rules. Your assertion that they can't simply make the roll against your prescribed DC is is what I take issue with. You may play it that way, of course, but doing so, IMO, puts an undo burden on a player who just wanted to make a nimble, athletic character that could jump, swim and climb better than others.