I appeared to have been mistaken. PerlIONext gets the next layer towards the bottom (system level), so PerlIOBase_close closes the layers starting at the top.

The problem actually lies in PerlIO::via.

IV
PerlIOVia_close(pTHX_ PerlIO * f)
{
    PerlIOVia *s = PerlIOSelf(f, PerlIOVia);
    IV code = PerlIOBase_close(aTHX_ f);
    SV *result =
        PerlIOVia_method(aTHX_ f, MYMethod(CLOSE), G_SCALAR, Nullsv);
    if (result && SvIV(result) != 0)
        code = SvIV(result);
    PerlIOBase(f)->flags &= ~(PERLIO_F_RDBUF | PERLIO_F_WRBUF);
    return code;
}
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It calls the rest of the chain (down to the system level) before calling your custom handler.

1. utf8
2. encoding(UTF-8)
3. via
4. perlio
5. unix
6. Mine ← !!

It should be calling your custom handler in order.

1. utf8
2. encoding(UTF-8)
3. via
4. Mine ←
5. perlio
6. unix

Reported as Perl RT#75780, including a trivial fix.