I was taught karate from the standpoint that karate was the "arm sword" (a la 1960's) and the idea that the sword shape for the arm is the strongest shape isn't peculiar to Aikido. Blocking and striking patterns of karate were taught to me that way as well, so while you and I have very many common understandings of martial arts, I probably see that commonality more from my beginnings rather than from art comparisons.
Judo taught me how to be centered, and to "make myself heavy" in order to execute techniques, but (of course) depended on leverages and kuzushi from mechanical movements more than from "centering". Throws like the side separation, etc. were clearly aiki-type throws, but still depended on the mechanical "devices" of Judo to execute.
What I noticed long ago was the "centered movement" of all martial arts at higher levels. "All masters move the same".
When my training partner used to attack his Judo teacher, he would say "my attacks must really suck"... His teacher would tell him, "No, you have great attacks... it made me move 2 or 3 inches..." I noticed that from all the master teachers I've trained with and under.
I watched Nishiyama Sensei at a clinic do a 270 degree turn, while defending against an attack, and it was like he was oiled in his movements. Everything just "slid past him". Toyoda Sensei was like that too... he would touch someone and their path would be redirected into the cleanest Aikido technique you can imagine. Kuniba Sensei moved likewise doing karate and jujutsu with us... so while the grounding comes in different paths of training, it's still the same thing... "tomato" and "to-mah-to".
I teach students that "karate is a force delivery art", "Aikido is an art of redirection and blending", and jujutsu is a "locking and pinning art". While all of them have similar characteristics and techniques that "look like each other", the underlying philosophy of force dissipation is different in each one... so it has its own peculiar character, and how that "intent" is expressed is somewhat different.
A mechanical locking technique might not look as "smooth" as a tenkan entry into kote-gaeshi, but effectively do the same thing force wise. The underlying movement to accomplish it might look different, but hopefully, it is based on being centered and creating the correct angles to avoid being hit.
We had some visiting black belts from another Aikido/jujutsu school train with us one night, and expressed to my partner "they would take a punch to give a punch". He laughed out loud at them, and told them if they tried that with either one of us, they wouldn't ever get their punches off... and they didn't. My underlying principle of intent is to knock the other guy's lights out ASAP, and then go have a beer (although I had to stop that).
The saddest day of my life was when I found out I had developed an allergy to beer
Life just ain't fair...