I do think people are being somewhat OTT.
The kind of person who would try to use headphones to strangle someone is firstly extremely rare, secondly would probably have their own strangulation thing with them, and finally I reckon with most headphones the wire would break.
- if you are still worried about them get some wireless ones.
To still hear around you, have the volume on low and only one ear piece in at a time.
I would always take a phone with me, the chances of someone trying to mug you when running are extremely low (even more so since would be muggers would know a runner can a- run from them, b- run after them and c- is unlikely to have anything else of any value on them). The chances of you falling and hurting yourself and needing to call for help are much higher.
Clothing wise - it just needs to be practical and comfortable - I highly doubt how fitted or not it is makes a difference to the extremely rare sick individuals that would actually do anything - however clothes/footwear that might make you trip or twist your ankle are dangerous.
If you need to be on roads high Viz so cars can see you - regardless of time of day and/or light conditions.
Basically though, if you'd be willing to walk there, then there is no problem running the same route.