I would understand the noun (which you probably won't find in a dictionary) to mean uncomplicated, pragmatic, willing to roll up one's sleeves and get to work. And seeing it refers to Bruce - man of the people, at ease in casual dress.
Most associations that occur to me are positive. And it's not annoying cheesiness.

hemdsärmelige Herangehensweise - pragmatic, non-formal way of approach

Yes, in general it is meant in a positive way. But when used together with 'zu' ('das ist aber [etwas] zu hemdsärmelig') then some mandatory formalism was skipped.

The manager used to treat such issues in rather a cavalier fashion.

Hier wäre ein Beispiel für die negative Konnotation der Hemdsärmeligkeit. Aber wo ist eine 1:1-Entsprechung in Form eines Substantivs? Weiß leider auch keine...

leave it to the French!

"Down to earth" may not capture the full meaning of "hemdsärmelig" but certainly comes close. Can you even turn it into a noun and be understood? - I leave that to the native speakers.

The German "hemdsärmelig" may have a more negative (unorganised and/or unprepared) touch to it than "play by ear" (which implies the same situation but assumes things will work out eventually and suggests a more relaxed state of mind while succeeding).

Normally I'd translate Hemdsärmeligkeit into "roughness", in the given case with Bruce Springsteen I think we need a more positive idea, so "casualness" might do, but maybe that sounds too formal here.

Sleeveless bravado

??