sysroot and arm-gnu-toolchain-10.3.rel1-x86_64-arm-none-linux-gnueabihf don't work

I'm using arm-gnu-toolchain-10.3.rel1-x86_64-arm-none-linux-gnueabihf.tar.xz to cross-compile
from host LSW running Ubunti 22.04 and target BBB running Debian Bullseye.

To cross-compile I must use sysroot, but it don't work. Running a simple hello world,

/usr/local/arm/gcc10.3/bin/arm-none-linux-gnueabihf-g++ -v --sysroot=/home/krister/bbb-sysroot hello.c -o hello10
yields:

/home/krister/bbb-sysroot/usr/include/stdio.h:27:10: fatal error: bits/libc-header-start.h: No such file or directory
27 | #include <bits/libc-header-start.h>

The reason for this is that only /home/krister/bbb-sysroot/usr/include is added.
Correct would be:
/home/krister/bbb-sysroot/usr/include/arm-linux-gnueabihf
/home/krister/bbb-sysroot/usr/include

Adding the path my self:
/usr/local/arm/gcc10.3/bin/arm-none-linux-gnueabihf-g++ -v --sysroot=/home/krister/bbb-sysroot hello.c -o hello10 -I/home/krister/bbb-sysroot/usr/include/arm-linux-gnueabihf

yields:
/usr/local/arm/gcc10.3/bin/../lib/gcc/arm-none-linux-gnueabihf/10.3.1/../../../../arm-none-linux-gnueabihf/bin/ld:
cannot find crt1.o: No such file or directory
/usr/local/arm/gcc10.3/bin/../lib/gcc/arm-none-linux-gnueabihf/10.3.1/../../../../arm-none-linux-gnueabihf/bin/ld:
cannot find crti.o: No such file or directory

The reason for this is that crt1.o and crti.o is not prefixed with the path:

/usr/local/arm/gcc10.3/bin/../libexec/gcc/arm-none-linux-gnueabihf/10.3.1/collect2 -plugin /usr/local/arm/gcc10.3/bin/../libexec/gcc/arm-none-linux-gnueabihf/10.3.1/liblto_plugin.so -plugin-opt=/usr/local/arm/gcc10.3/bin/../libexec/gcc/arm-none-linux-gnueabihf/10.3.1/lto-wrapper -plugin-opt=-fresolution=/tmp/ccnejzOj.res -plugin-opt=-pass-through=-lgcc_s -plugin-opt=-pass-through=-lgcc -plugin-opt=-pass-through=-lc -plugin-opt=-pass-through=-lgcc_s -plugin-opt=-pass-through=-lgcc --sysroot=/home/krister/bbb-sysroot --eh-frame-hdr -dynamic-linker /lib/ld-linux-armhf.so.3 -X -m armelf_linux_eabi -o hello10 crt1.o crti.o
/usr/local/arm/gcc10.3/bin/../lib/gcc/arm-none-linux-gnueabihf/10.3.1/crtbegin.o -L/usr/local/arm/gcc10.3/bin/../lib/gcc/arm-none-linux-gnueabihf/10.3.1 -L/usr/local/arm/gcc10.3/bin/../lib/gcc -L/usr/local/arm/gcc10.3/bin/../lib/gcc/arm-none-linux-gnueabihf/10.3.1/../../../../arm-none-linux-gnueabihf/lib -L/home/krister/bbb-sysroot/lib -L/home/krister/bbb-sysroot/usr/lib /tmp/ccpVc2CC.o -lstdc++ -lm -lgcc_s -lgcc -lc -lgcc_s -lgcc /usr/local/arm/gcc10.3/bin/../lib/gcc/arm-none-linux-gnueabihf/10.3.1/crtend.o crtn.o

correct would be:
/home/krister/bbb-sysroot/usr/lib/arm-linux-gnueabihf/crt1.o
/home/krister/bbb-sysroot/usr/lib/arm-linux-gnueabihf/crti.o

Is this a bug in the compiler or is it just me ?
Is there anyway to get around this ?

The latest 12.3 have the same problem.