I haven't really kept track of Android sites, since I'm rarely playing native Android games other than The Battle of Polytopia, an accessible, but endlessly replayable Civ-clone. Please send help - I've been playing it almost since its release seven years ago and can not stop. The base game is free and each additional tribe is only a small one-time purchase. The more tribes you have purchased, the larger maps are unlocked. A single tribe purchase unlocks multiplayer. There's also one additional skin for each tribe. Perhaps the best monetization model on mobile to date.

There are few standout titles like these for mobile devices. If you want to seriously play games on your phone or tablet, use an emulator. I'm currently revisiting Vice City in GTA Vice City Stories, a lesser known, but nonetheless high quality spin-off in the series (there's also one for GTA 3: Liberty City Stories), which runs perfectly in AetherSX2 (although I should switch to NetherSX2, which continues development of this PS2 emulator). If you have a less powerful device, you can play the only slightly inferior PSP versions (worse textures and significantly downgraded lighting, some missing side-missions, but overall the exact same games) in PPSSPP, which has much lower hardware requirements. You can even add a second thumb stick to the PSP versions using this patch:

https://github.com/Freakler/ppsspp-GTARemastered

The PSP version of Chinatown wars is superior to the Android port, having better visuals and more missions, so consider playing this one in PPSSPP as well.

I've recommended PSP racing games here:

https://beehaw.org/comment/2784912

And PS2/GC/Wii action games here:

https://beehaw.org/comment/2994175

I would highly recommend using a controller for most of these titles. You can connect almost anything to an Android device, either wirelessly or through USB. Wii or newer, PS3 or newer and Xbox One S or newer work via Bluetooth in every emulator I've tried. With wired controllers, you can go as far back as you want. I'm playing PS2 games using an original PS2 controller with a PS2 to USB adapter, which is connected to a USB A to USB C adapter. If portability is of concern, 8bitdo makes very high quality, but nonetheless affordable controllers that you can easily fit into even the smallest pockets. The d-pad on my FC30 Pro is perhaps the single best d-pad I've ever had the pleasure of using.