Bike shoes and clipless pedals do make a difference for me. On my commuter bike I have power grips pedal straps which - random guess here - 85% of the benefit while still allowing you to wear regular shoes Pedal Kits ? Power Grips

If you're changing your shoes when you get to work anyway, then sure, go clipless.

A big deal for me is the amount of cargo I'm carrying, and hence what kind of bike I am using.
On my commuter/tourer road bike, with panniers, I do my 12.5 mile commute in about 48 minutes.
I can do the same trip in about 39 minutes on my triathlon bike. But my tri bike doesn't allow carrying anything other than water, emergency kit, and a gel or bar.
Also, those numbers are for actual riding time. When you throw in stops and waiting and passing slower bike/pedestrian traffic, the difference is smaller.
And what's 11 minutes x two trips in a day anyway?

It's going to be hard to make your commute shorter by a difference that will really matter without going e-bike.