A few things from a SAHM who works part time while her extremely active toddler naps (toddler is advanced on milestones, so I know how rough a very physically active young one under 2 can be). I do all the household chores (cooking, cleaning, errands, etc) with her in tow so I've had to get better about learning how to get stuff done while also simultaneously being the only person responsible for supervising her.
1. Train your kiddo to be able to be alone for a half hour in their crib as needed. Put the kid in the crib with their favorite, safe toys and books, and go take your shower. There is nothing wrong with letting a young toddler sit and play in a safe area for 20 minutes while you get shit done. And if they cry or fuss? So what? You aren’t disappearing for an hour, you are taking a 15 minute shower. This option works best for activities that are short in duration (showers, important phone calls, etc) but where you can't actually supervise the child adequately.
2. Enforce naptime/quiet time – if your kid wakes up from naptime but is happily rolling around and talking to themselves, why jump up and go grab them? obviously don’t leave them in there for hours, but like point #1 – if the kid is fine being alone for a little longer, don’t shoot yourself in the foot and purposefully forgo such a great opportunity.
3. Be smart about your choices when you do have free time – like you have discovered, lounging while the kid naps is a surefire way to get nothing done. I use daily to-do lists of what I need to get done during naptimes, and I try my best to complete it. Sure, that means I don’t have down time during the day, but the upshot is that I don’t have to do all this stuff at night when I’m even more exhausted. Prioritize the activities that are least likely to be achievable when the kid is awake. So for example if you need to fix the kitchen door and need 2 people to do it, that is a perfect task to do while baby is napping.
4. Learn how to operate around the kid rather than wait to do everything while they are sleeping – toddlers need a lot of supervision, but they do not need 2 parents just hovering over them staring at them all the time. Baby gates are your friend. Gate off your kitchen while you cook and leave pots, pans, boxes, and toys all over the floor. You can cook, do dishes, clean up, etc while kiddo plays. Similarly, gate off and baby proof your living room so you can lock the kid in there while you run and grab the laundry. Yes this means you will have to leave the baby unsupervised for 60 seconds. No they won’t die, I promise.