There's no getting around the control issues without a clunky peripheral, but this site helps to find the rare examples which aren't designed to bleed you dry at least: https://nobsgames.stavros.io
Predictably many of the high quality games in that category are ones which started their life on PC or consoles and were ported to mobile later. Mobile-first game development is rotten to the core.
The Nintendo DS light fits comfortably in a pocket on the other hand and had great games made for it that took advantage of that platform's featureset (I personally think the legend of zelda games for that platform have some of the best controls out of any handheld nintendo game with the stylus gestures). Clunky peripheral is from a lack of imagination not from a technological impossibility.
I remember when Sony released the Xperia Play phone a decade ago, with a built in slide out controller. Something like that would be great to have, but maybe instead as a peripheral that clips onto the back on your phone. Wouldn't be much thicker than the battery cases that are common now.
Not really. Apple could design a pretty clean peripheral if they wanted to and it probably would be about as thick as the nds was. The iPad keyboard is a good model of how that might look, secured on a magnetic attachment perhaps able to be folded into different orientations. Like I said, little imagination however has taken place in this sector. Apple is content to take their 30% cut off the whale gamblers. The iOS game market is certainly massive enough where that zero overhead rent seeking on the part of Apple is quite lucrative, apparently in 2022 almost $50 billion were spent on ios games alone. $15 billion to Apple thanks to their pizzo.
Apple Arcade, a modest subscription in itself, is filled with mostly kid friendly games and none of them are allowed to have iAPs (or the iAPs are made free and then bundled in order to be featured in the Arcade catalog).
I would suggest getting a last gen handheld, since they're pretty cheap now, have good libraries of actual proper games (especially with emulation), and physical controls just can't be beat. I got a near-perfect vita from Japan for $90 and I'm quite happy with it. Not as convenient as the phone in your pocket, but if you're carrying around a backpack anyways it's not bad, and for kids maybe encouraging them off smartphones ain't so bad (:
Predictably many of the high quality games in that category are ones which started their life on PC or consoles and were ported to mobile later. Mobile-first game development is rotten to the core.