Not true. An obit is not an infinite "plain" with a finite "hill" on but rather a finite "valley". The ball will exit the valley on the other side unless it loses excess kinetic energy somewhere in the valley.
Anything that approaches the sun will do so with faster than escape velocity because the gravitational potential energy gets converted into kinetic during the approach. Newtonian mechanics is time-reversible - just like it's impossible for an object in orbit to spontaneously escape without gaining energy from somewhere first, it's impossible to enter orbit without losing energy to another body.
Small nit: this is for objects approaching from infinity in a two body system. Otherwise the object can be in orbit and already be “captured” there. For example, the moon also approaches earth slightly over the month while speeding up, then slows down while moving away. Or it could just be a 3 body system, which is chaotic and can’t be modeled accurately and can have objects spontaneously eject from the stable system (even though the physics is indeed reversible)
Your first nitpick only makes sense with a weird definition of "approaches" for the context, so I think it just adds confusion. They're pretty clearly talking about something that is headed in the direction of the sun and not already in orbit.
And the comment you're responding to already mentioned that other bodies can make a capture happen. Nobody was saying capture is impossible.
I thought the context is orbital mechanics, for which the term approach is the common term for a portion of the trajectory where the bodies move towards each other. Take any Wikipedia article for an orbiting solar body and search approach eg: https://en.m.wikipedia.org/wiki/Halley%27s_Comet
That is not true. If it approaches with less than escape velocity it will gain all the velocity necessary to escape in the process of approaching the Sun.
You could think of it as speeding up as it falls toward the sun, it then slows down by the exact same amount as it leaves the Sun.
In order to stay near the sun it needs to lose some of that speed, and given that momentum is conserved, the only possible way is to either hit the Sun or send that momentum to a third object.
