With your latest sources (13 Oct 2022), I find that with some compilers the run for the LSODPK example produces many warnings with the pattern

DLSODPK- Warning. Poor iterative algorithm performance seen
at T = R1 by linear convergence failure rate = R2
In above, R1 = 0.1377169669546D+01 R2 = 0.9166666865349D+00

Here are some specific details, all on a Windows 11-64 PC.

S:\ALGO\ODEPACK\UJ2\odepack-master\src>ifort /O2 /c M_odepack.f90
S:\ALGO\ODEPACK\UJ2\odepack-master\example>ifort /O2 /I..\src lsodpk.f90 ..\src\M_odepack.obj
S:\ALGO\ODEPACK\UJ2\odepack-master\example>del ccout & lsodpk

generates 10 sets of warnings

If I use the Ifx compiler instead of Ifort, no warnings. If I use Gfortran 11.3 (Cygwin64), with -fallow-argument-mismatch -O2, no warnings. If I use the NAG 7.1 compiler with -dusty -O2, I get the warnings.

My hunch is that this sensitivity of the algorithm to slight variations in floating point calculations is inherent in the algorithms used, rather than being caused by bugs in the source code or the compilers.