I have to admit I'm stumped.
I almost never use accelerators or 'gel' types...usually just 'grocery store' Super Glue...and your description sounds pretty much the same as the way I've always used it. That having been said, I've rarely had the difficulties you describe. I always try to use the absolute minimum amount -- at least for the initial bond -- then go back and 'reinforce' if necessary.
The only thing I can think to ask is whether you work in a low-humidity environment, since CA needs airborne moisture to set and cure.
Since you said PE is your main use, do you clean your PE with alcohol or something similar first? It sounds almost like you've got oil or something on the parts. (Apologies if it's stating the bleeding obvious, but I'm struggling here.)
The 'not taking hold' part of your original description seems odd to me. I would not suggest applying the glue to both surfaces, as that would offer no real advantage. Those who use accelerators regularly sometimes suggest applying the glue to one surface and the accelerator to the other -- might be worth trying, if you haven't already done so.
The 'good' news is that that applicator shouldn't make any difference, as long as it's not putting on too large a quantity.
Doesn't seem like much help...but I've got no better ideas. Maybe I've missed something obvious that somebody else will pick up on.