I still do it the old fashion way, except I use ECHO BOY as the tape deck. The "trick" to not muddying up the mix is to PAN the effect exactly on top of the instrument track you are trying to effect and to use a delay time in somewhat in sync with the tempo of the song. There is a reason for this in sync delay time, that is, too hide any slapback sound that might slip through.
The other OLD SCHOOL process was to send the delayed signal direct to the reverb buss and have ZERO effect without reverb in this behind the scenes effect.
This entire process has been described as the REVOX tape trick used on many hit records from the early 70s. They used to create a physical loop of tape of about 25 feet and hang in from the ceiling and walls to create a giant loop of tape for this big ECHOPLEX machine. So, another thing to keep in mind is that both the right and left sides of your delay need to be set at the same time so that the effected track return stays in MONO so it can sit right on top of the original track. (you can control the PAN placement of the effect using the SEND pans to this special delay buss). In the old days, the only way to control the delay time for this effect was by changing the speed of the tape between the record and playback head, which normally would be fixed to the IPS speed, but these old REVOX decks had a variable capstan speed control to tune in the delay time to the tempo of the song.
There is no doubt that when done correctly, you can not hear the delays but turning off the returns of the delay, makes your track seem muffled.
I was generously taught this REVOX trick from a somewhat famous english engineer when I asked him how such and such a sound from a certain album was created. It took me over a year to finally work out what he told me they use to do. At the time, I never heard of the HAAS effect but have learned from peeps on the internet that it now has a better name than the "REVOX TAPE trick".
Now, if someone out there is more familiar with this technique and can shed some more light on how to better accomplish this with a modern DAW, I am all ears.
The piece that took me the longest to get was that the effect had to sit right behind the original tracks panning and that the only way to do this was with a stereo delay and that both the RIGHT and LEFT delay times, had to be the same.
Jim