Cue rant from me. It is actually quite DIFFICULT to do your shopping my bike in the UK.
I have a car in an underground car park. Fetch coat, mask and keys. Blip remote control, car unlocks. Drive to supermarket. Park in a dedicated space. Car deadlocks. Take trolley and fill up. Unlock boot, transfer groceries. Return trolley to trolley point. These days car locks are effective and you dont expect your car to be stolen or vandalised.
For the bike I put on SPD shoes and maybe special clothing. I have to open a bike cage then unhook my bike from hanging on the wall. Undo heavy chains. Hook on the panniers.
I park on racks right outside the supermarket but I still have a fear of bike being interfered with or stolen in broad daylight.
Panniers into a trolley and fill them at the checkout. Panniers back onto bike. On return home I have the faff of locking the bike up securely with heavy chains.
And I'm lucky enough to have a dedicated bike cage in a dry car park.
When i lived in .UK, my shopping by bike was:
Put on normal shoes, and a some clothes, (Sainsbury's Canterbury being sticklers for fully clothed shoppers it seems), pick up Brompton and backpack, walk down stairs, unfold Brompton, cycle to shop, fold up Brompton, place Brompton and the backpack in the trolley. Wander round picking up stuff, goto checkout, pay, pack it all into backpack, place backpack on back, wander outside, remove Brompton from and return trolley, unfold Brompton, ride home, fold Brompton, climb stairs, take backpack into kitchen, put stuff away.
Your issue is you're using the wrong bike, you've got the wrong pedals, and some barbarian has designed inappropriate bike storage...
Here in .NL I do much the same as I did in the UK, except there's a lift in there, and I unfold my bike at the top of the stairs, not the bottom. A lot of Dutch people park their bikes just outside, and leave the empty panniers attached. They then pack shopping into normal carrier bags and put those in the panniers.
It's no secret that I'm not a big fan of SPD's, but why you would need them for shopping I've no idea.
J