I completed both Puzzle League and Desert Island Escape with no amiibo figures, so it can be done.
For Puzzle League, just keep at it. It's partly skill and a lot of luck. The horizontal wipeout is helpful so use it when you need it and try to get the most out of every use by taking out full lines and trying to plan combos if you can. I've played puzzle games for what seems like forever, and some of the levels were still frustrating for me. If you're doing poorly and find yourself getting frustrated, just take a break from it and come back when you're recharged. Eventually, you'll get a random string of luck to help you through.
For Desert Island Escape, I only had access to my current villagers to get me through, but I managed. Luckily, I had Sydney the koala and Scoot the duck who were both very good. Koalas have Whimsy, which lets you move 7 spaces per turn or 0 based on a spin, and they're good at combat and getting honey. It takes some luck because of the spin to move aspect, but it's worth it for the most part. Ducks have soft-footed, which means they can sneak past enemies sometimes, and they can move a respectable 5 spaces. My third member was Monty the monkey because he could move 4 spaces, which was the next best option I had available to me. He worked out well enough and was a solid team member.
If I was having a bad run of luck with Whimsy or if I ran into a map with a lot of mountains, I sometimes switched out Sydney for Hans. Hans had the Strong & Fearless ability so he was able to smash through mountains (earning a few extra points at the same time) and also did really well in combat. The drawback was that he only moved 3 spaces.
I set up my team so that the food rations required only 3 or 4 rations per day. My best combatants were set up to go first so they could take out any threats I ran into and then the rest of my team members focused on collecting items and rations or just expanding the map for more points if there was nothing else within their range.
A lot of Desert Island Escape was based on memorizing the maps. I tried to beat every level on the first go if I could, but along the way I memorized where the enemies and items were usually located. They rarely changed position. Then, I would re-run the island back to back if possible, while my memory was still fresh and try to get a high enough score. I didn't move on to the next map until I had completed the map I was on with the score that I wanted.
For the tools, I focused on making the slingshot first to help with combat. Your score for defeating enemies goes up incrementally with each sequential win, so it's a good way to get points if you can win consistently. Then, I would try for the net if it was available to get honey to build my rations for a better score at the end. If I didn't like my odds, I would try to avoid combat or bee stings, though, since losing on either of those would make me skip a turn on that character. It's better to be able to move every day. I never really bothered with the shovel except for one level that I think was very heavy on the pitfalls. I also didn't bother with the fishing rod unless I could make a gold rod that I would use if I felt pretty sure I knew where a log or sail was to save me a day of travel, because you can cast it and pick up the item with the gold rod. But it breaks after one use, so it's not worth it for fishing.
Anyway, those were my strategies working with what I had available to me and I got through it all eventually. I hope some of this helps you out.
