Actually, almost all cameras today still have shutters, including P&S...the only difference is whether it's a mechanical shutter or an electronic one. Depending on the P&S model, some actually do have mechanical shutters - usually the more advanced models...with those, there usually is some way to still tell what the shutter count is. Many cameras still store shutter count within the EXIF of the shot - but you need to use a deeper EXIF reader to get into the extended EXIF settings - one of which is usually the shutter count (Nikon, Pentax). Some cameras have a combination of keys you can press to get the shutter count (Olympus, some Panasonic). Others require a third-party software that you can use to get the shutter count from an original unresized photo (Sony, Canon).
For cameras with electronic shutters, it's up to the manufacturer to decide if that count will be kept somewhere - some may, some may not. It's generally considered unnecessary, since the main point of tracking shutter count was to get an idea of how much life may be remaining on a mechanical shutter that can eventually wear or fail. With electronic shutters, failure can occur anytime and regardless of the number of times it's been shot. I would still try some of the shutter count EXIF readers for that model of camera, to see if the information is still stored the same way in that camera's EXIF.