When he was about 20, Yehudi Menuhin,
one of the best concert violinists of
the 20th century, concluded that suddenly
he didn't know how to play violin.

Why? Because before his teachers, back
to his age 7 or so, had carefully walked
him through the music note by note and
bar by bar and, likely then, the artistic
expression. But, still he was darned
gifted. So, a few years later, relearning
violin, he was fine again for the rest of his
life.

Being able to do arithmetic, algebra, and other
mathematical operations in your head is fine.
E.g., during WWII, John von Neumann, one of the
best mathematicians of the 20th or any century,
sometimes went to people working on war projects
to see if he could help them out. At one stop,
a guy was unsure about how to find a solution,
so von Neumann suggested, "Let's try a local
solution via infinite series and look at a few
terms. So, von Neumann did the arithmetic in
his head [he could take two eight digit numbers
and multiply them together in his head]. When
von Neumann left, the guy was near tears. Why?
Because he had been up all night using a mechanical
desk calculator to get just some of that arithmetic
right!

I found in doing research in math (I've published
such stuff), once get into the problem well then
do most of the thinking for the actual research
just between the ears and writing very little.
Can even do some of the algebraic derivations
just between the ears. The actual research, at
least the way I do it, is heavily 'conceptual'
where writing isn't very efficient anyway.

From what you wrote, I see nothing wrong with your
abilities. If you are not handy with paper and
pencil, then it is not too much to expect you to
become handy with just some usage. So, try, and
soon you will 'get it'. It sounds like your
problem with calculus was mostly just a problem
with paper and pencil -- so, 'get it' with paper
and pencil. Unless are, say, another von Neumann,
can't expect to do significant derivations in
calculus, say, integration by parts or multiple
integrals, without writing down the expressions
step by step.

But, of course, for calculus, long ago some software
efforts for integration worked out what expressions
could be integrated in closed form and what ones
could not and wrote software to do the integrations
for all the ones that could. The results are in
at least one of the old computer algebra packages!
So, with that software, "Look, Ma, no pencil or
paper!".

About your suspected faults, don't be the least
discouraged about them! Some are just your
imagination. Some can be easily enough corrected.
Essentially all the rest can be circumvented.

In particular, it appears that you are concerned
about some skills that were supposed to take
you from a few days to at most a few months
to get good enough with way back long before
you shaved! Doing catch up on that material
now should be a few days' walk in the park.
E.g., in K-12, I didn't learn English grammar
and punctuation very well. In college
I was in math and physics where this gap was
not very important, and otherwise I just
wrote only sentences where I did understand
the grammar and punctuation. Now I've long
had a 12th grade English grammar book right
at hand and look up details when I'm in
doubt; long ago I got nearly everything I
missed in K-12.

Back to working without writing, in plane geometry
I refused to do the homework the way the teacher
wanted, write out in full detail proofs of just
three problems. Instead I worked all the non-trivial
problems and then also the more challenging ones
in the back of the book, often writing nothing,
sometimes writing in the margin of the book,
sometimes on scraps of paper, and only for a few
of the most difficult problems in the book, actually
writing out a complete proof. Worked fine! When
the teacher wanted to see my homework, I showed
her nothing, and she was torqued. When I came in
second in the class on the state test, she was
'confused' since she thought that I didn't do any
of the homework! Heck, I solved every non trivial
problem in the book and, thus, likely did the
most homework of anyone in the class. Also, I
slept in class -- I learned from the book in
a quiet room, not when she was making noise
in front of the class! Being able to learn
math just from a book proved crucial, the main
way I got my Ph.D.!

Lesson: You don't always have to do it their way!