Fortunately I also have a Mega STe (at present in the midst of being repaired unfortunately) upgraded to accept HD floppy drives (IIRC it was a matter of replacing a GAL chip and soldering a wire on the back of the motherboard), so that's a nice way to check if the drive is at fault.
I just checked the drives and found out the Mega STe has a Sony MPF-520-1 inside while the STe has a Sony MPF-920-E. I believe I modified them both in the same way:a)
change the drive from "Drive 1" to "Drive 0" (they used to have jumpers for this but later they just soldered zero-ohm SMD "resistors" in place which you have to desolder and re-use in the new position or just use a short wire)b)
solder a wire across the "CTIN" disk detect switch (for a permanent connection regardless) to fix the "disk change" issue (so the computer won't think the same disk is inserted when in fact another floppy has been inserted)c)
connect a wire between pin 2 (the 34-pin Shugart connector) and to the "HDIN" (High Density disk detect) switch (the switch-pin which doesn't go to GND)d)
the "High Density detect" signal is further taken from the same connection as in step c (above): pins 2 of the 34-pin connector/one side of the "HDIN" marked micro-switch. A wire from here goes to the 1.44MB/720KB HD-module board (Elco or whatever)e)
insert the 34-pin cable from the Atari into the drive, but upside down! In other words, the way it won't fit (because there's a notch in the connector that doesn't match), so you have to cut away the notch to fit it "the wrong way". It's probably a good idea to mark both the disk drive connector and the connector at the end of the cable with a blob of white paint or whatever so you won't be in doubt if you ever need to disconnect/reconnect it
Aesthetically I've also solved the disk eject button (I didn't want to butcher the outward appearance of either computer) by removing the front bezel, taking the (rectangular) PC style eject button that came with the drive away, cutting off part of the metal lug it was mounted on to allow the Atari style eject button I had from my previous (DD only) drive which came with the machine (I think it was a Sony as well). A little blob of glue helped keep it fastened to that metal lug, then remount the plastic bezel.
You've probably modified the drive similarly but I thought I'd just mention it.
When I get the Mega STe it should be interested to try both drives out there first (I can't remember if I tried them both out in that computer with HD floppies), before attempting to get one of them to work in the STe with the Elco or another HD module if it turns out to be more trouble than it's worth.