There is a saying in one of the old schools of jujutsu (that has a lot of suitemi) that goes along the lines of
"If you are entangled with your opponent, then the only way to throw them is to suddenly perform a sacrifice throw"
It's worth noting that the sacrifice throws from older styles are very damaging to your opponent & either end with you in top control (usually mount) or with a lot of space between you and your (damaged) opponent
This is where the overlap between old jujitsu and old Te ends. Te shares the do it very quick aspect, but the Te approach is to stun with precision. Either using open hands to the throat or eyes, or elbows to the temple, jaw or carotid.
Atemi waza as judo would call it. Kano emphasized its importance as did other branches of jujitsu.
i think the philosophical difference between te and jujitsu is one prefers to go to ground fighting as quickly as possible, with the best or most dominant position taken.
The other seeks to stun and overwhelm, and send the opponent to the ground without deliberately turning it into a newaza fight. Where a higher skilled party might easily take the fight.
I am reminded of an old event that happened between Gichin Funakoshi and a judoka who challenged him. Gichin begged off, but the young judoka wouldn't let it go. The challenge match was eventually accepted along the road. The judoka closed in and just as he was getting his grips in, GF threw a hooked punch to the solar plexus with a frightening amout of power.
The judoka had his diaphragm bruised and it was spasming so bad that his breathing was interrupted for close to twenty minutes and he turned blue.
GF was very worried that he had killed the young man.
It was after this that GF started refusing all challenge matches, because he didn't want anyone to die at his own hands.
Much earlier in life, GF was tossed three times by Choki Motobu with a kotegoshi. Choki thought striking GF in front of GFs students would be doing too much harm to the honor and dignity of GF, and was beyond the lesson he wanted to teach him.
Interesting that a seasoned master level karateka could throw another master easier then a judoka could... and without the atemi waza.