OS 4.x runs on PowerPC processors, and there are a bunch of dedicated systems like the X1000, X5000 or SAM460EP. You can also run a "classic" OS 4.x version in emulators or on classic Amigas fitted with PowerPC accelerators. You could always snag a used Mac Mini PPC and run MorphOS on it for a similar experience; MorphOS is a sort of "off-shoot" from the Amiga family tree but has very active development including up to date browsers like https://wayfarer.icu.

Hyperion have also released updated and bug-fixed versions of "classic" AmigaOS, and you can purchase the ROM sets and disks for e.g. OS 3.2 to use with an original machine like your A500.

For HDMI output, I use an Indivision card. It's a little fiddly to fit (has to clip in over the top of some other chips) but gives a great stable and crystal-clear DVI or HDMI output. There are also external "flicker fixer" boxes or scandoublers that turn the Amiga 15KHz video output into something useable by modern displays, but in my experience they tend not to give as good quality output or have lag issues when playing games.