I have the ASA flight timer you mentioned. It's got local time (both 12/24 hour) and UTC, six independent approach timers plus a separate stopwatch, a notepad, backlight, runs on a single AA battery. Everything you need.
But it's also big. No way you can easily mount this somewhere portable. Not on your wrist, not on the yoke/stick, not on a lanyard around your neck, not on your kneeboard. The only way it's really useable would be if you mount it permanently in your cockpit somewhere. Today, it mostly lives in my flightbag and I honestly don't know why I'm still carrying it around. I don't think I've ever used it in anger so far.
For the PPL exam, all you really need is a clock of some sort. For me, a simple analog watch did just fine but hey, go wild and get a digital watch if you think it'll make you a better pilot. But remember that the accuracy required for the PPL navex is only along the lines of +/- 5 minutes ETA. Any simple clock will do.
Only once you start doing the CPL and/or IR, the accuracy requirements go up, and you need something that you can start/stop. But even then an analog stopwatch will probably do just fine.