The end chase bit requires to press the "use item" button after picking up the axe, though I have heard from others that for some reason they can't do anything at all after picking it up, I'm not sure the exact reason.
The punishment for getting caught being so harsh was to add tension the further you get in, though I can definitely understand it becoming more frustrating than engaging, especially considering how it's not super clear what you're supposed to do after picking up the axe. I'll definitely take that into consideration going forward, if I ever decide to do a sequel or remake.
Thanks for the feedback!