F15 & gear (B737) or F5 & speed brake will give 1200-1500fpm. No drama and the G/S can be captured before 4nm if you begin the process soon enough. The secret is to realise what is happening, either caused by misjudgement or ATC vectors, and act promptly before the hairs on back of neck get prickly.
This was originally an innocent technical query on my part, but your words now seem to take on more significance when I read the thread on the Turkish Airlines accident.
It would appear the a/c was intercepting the G/S from above at less than 5NM much as you describe. The rest remains conjecture.