> 1. I've set my alarm for 8am local time every day. I don't want my alarm clock to go off at 3am just because I'm in a different country
Without timezone, your clock would actually go off in the middle of the night when you are in a different country. Your clock does not know you change location (no timezone, no location-dependent time), so it will alarm you after exactly 24 hours since the last alarm, which, in another country, can be midnight.
> 2. If I have an appointment at 10am next November, that's actually a datetime with a locale (given that my appointment is at 10am local time, no matter whether the DST rules change between now and then).
If everyone is staying in your city, then that works great (sans catastrophic continental riff or meteorite strike). When at any point you involve more than one geographic location, then the appointment is meaningless without timezones.
Now, maybe it is actually the case that you and your software only care about one geographic location in perpetuity, but just like assuming 32-bit IP addresses, 8-bit ASCII, and length-oblivious arrays are all you need, the teeth marks from its consequential bite tend to be quite deep and sanguine.