I'm left-handed but have no problem peeling potatoes. I use an ancient peeler I inherited from either my MIL (left-handed) or my Granny (right-handed), can't remember who it was now, but it works perfectly for me. I'll be bereft when it finally gives up the ghost. Picture attached of a very similar one, and also a picture of something very like my potato masher. This is how my Mum mashed potatoes and I've always done it the same way.
I peel the potatoes (King Edwards usually - they have to be floury, not waxy), cut into chunks of even size, but not too small, put in a pan with cold water and a bit of salt, bring to the boil, turn down the heat so they don't boil over, partially cover pan with lid, boil for a few minutes until cooked through to the middle. Drain off the water, put back on ring with a tea towel over to absorb the steam, then put in a good big bit of butter and a little milk. Mash till smooth, which takes seconds. Can add mustard, sauted leeks etc to add interest. Mashed potatoes mixed with grated cheese is food of the gods!
It possibly helps that I have a dishwasher but I don't find washing the pan and the masher tricky on the odd occasion I've done it by hand.
It's not that quick and it does require a bit of attention, but I love mash, so am happy to make the effort when I fancy it.