Fabric softener is a scam

Edited: spelling corrections

Regular laundry detergent leaves residue as well, that's why a lot of washers have self clean cycles.

Vinegar will cut most residues from a washer. It's one of the reasons it works as a softener, it removes detergent residue and minerals from hard water to make your clothes soft. As a side note a lot fabric softeners does the same thing by coating this fibres of the clothes in refined beef fat.

If you want a really clean machine (specific to most HE washers) put half a teaspoon of TSP(trisodium phosphate) in the empty drum of the machine and another half teaspoon where you put your detergent, fill the bleach cup and the fabric softener cup with vinegar. Set the machine to run a pre wash cycle then a heavy duty cycle on the hottest water temp(if there is a hot tap that is after your clothes washer in the pipeline, you can run that tap until it gets hot before starting the washer. HE washers use a lot less water so most of the time they are filling with cold water before the hot water rescues the machine) and as many extra rinses as your machine will allow.

After the TSP removes most of the residue, it's rinsed away and then the bleach cup of vinigar is released with the next rinse cycle. Since TSP is basic and the vinegar is acidic they react and the TSP is neutralized, this may lead to some deposits of various salts(depending on the exact reaction ratio, if there is an excess of vinigar then the acidic solution will help prevent the salts from depositing.

When the fabric softener cup of vinigar is released in the next rinse we ensure that all acidic solution is flushed through the machine. This again will help with any salt deposits and help clean any of the remaining residue that reacts better to the acidic solution.

Any extra rinses after that help make sure that anything that can be washed away will be. Your washer will be very clean inside.

TSP can be purchased at the hardware store for cleaning surfaces before you paint them, just make sure it's actually TSP and not a TSP replacement. Half a teaspoon of TSP in with your dish soap will make you love your dishwasher again. A little goes a long way, and using less TSP is more likely to give you a good result than using more.

That enormous rant aside, I have used homemade laundry soap recipes in several HE machines including two combination washer dryers(where the washing drum is the same as the drying drum, looks like a single HE machine, not a stacked washer and dryer) and in general those machines have more residue problems. I haven't really had any issues. That might be related to the fact that I do not put an excess of soap or detergent into the machine, again often less is more, and if you need better cleaning it's better to do a prewash cycle than add more detergent into the main cycle. More detergent =/= better cleaning, just more residue. Try washing a dish by hand with only liquid dish soap, no water, no damp sponge, just soap and a dry brush or dishcloth. While this is an extreme example, it does help illustrate that detergents or soaps allow grease and grime to be dissolved into water, and that's what allows it to clean effectively, the soap or detergent is just there to get stuff into the water.

Borax sensitivity is a legit concern.