I agree to an extent that COBOL was "indentation-sensitive", but it really wasn't "indentation-sensitive" but rather "column-sensitive". The original COBOL coding format had specific requirements of where source code elements had to begin or the compiler would spit it back out as an error.
- Column 1-6 was a sequence number
- Column 7 was an indicator
- Column 8-11 was "Area A" and certain elements had to start there
- Column 12-72 was "Area B" and certain elements had to start and be contained there
- Column 73 - 80 was a system generated number when printing - this area was ignored by the compiler
In modern COBOL (to the extent there is such a thing) there are other formats allowed that eliminate these requirements.
When programs were entered on punched cards (before my time) coding was done by paper and pencil first on coding forms that adhered to this layout - once it was laid out, you could get someone else to keypunch your program.

Column 1-6 was a sequence number - I think the original theory was that if you for example, dropped your card deck, it could use this to sort the cards (and your program) back into the correct order. In practice, this got used for a variety of purposes because the sequence number was not enforced. For example, we would use it to tag lines with a ticket number or version number for bug fixes.
Column 7 was used for indicators:
- * meant it was a comment
- - meant it was a continuation of the previous line
- / meant that when printing your code listing, it would page break
- D meant that the line would only compile if WITH DEBUGGING MODE is used
Column 8-11 was Area A and division, section, and paragraph names had to start there, as did things like FD (file declarations) and level numbers. If you tried to put a level number or paragraph name in column 12, it would cause a compiler error (or be recognized as something else). You could indent within the Area, but those elements had to start within there.
Column 12-72 was Area B and all code went here. If your code extended past column 72, that part would be ignored and likely cause a compiler error - or it could be the source of a bug if the line was syntactically correct without the part that was in column 73. (I've experienced this over the years.)
There was no indentation requirements within that area - your entire program could be entered there with no indents at all within the code.
So within Area B, it doesn't matter if I use:
PERFORM 0200-MOVE-DATA
VARYING SUB-VAR FROM 1 BY 1
UNTIL SUB-VAR > 20
or
PERFORM 0200-MOVE-DATA
VARYING SUB-VAR FROM 1 BY 1
UNTIL SUB-VAR > 20