The Amazon Echo works great as a bedside alarm clock, especially if you have an Echo Spot (which is pretty much targeted for that very purpose). However, if you're worried that your alarm won't go off when the Wi-Fi goes down, there's actually nothing to worry about.
Lifehacker recently published a PSA urging readers not to use their smart speakers (like the Echo and Google Home) as their alarm clock, because the author's friend once had an alarm that didn't go off the same morning the Wi-Fi went out.
There are many reasons this might have happened, but it probably wasn't because of the Wi-Fi---the Echo's alarms are designed to work even when Wi-Fi isn't available.
I tested this out myself using a regular Echo and an Echo Spot (which I use every morning as my alarm clock and it works great). I started off by setting an alarm using Alexa, then shut down my house's entire internet connection. I went to bed and woke up the next morning to my Echo Spot's alarm going off without a hitch.
I even tested it again by setting an alarm and telling Alexa to wake me up to some streaming Spotify music. Surely that would confuse my Echo and render the alarm catatonic without Wi-Fi, right?
It turns out, the Echo waits one minute to try and get a Wi-Fi connection to play the song, and if it can't, it will instead play the default alarm sound that the Echo came with when it shipped. So worst case scenario, you wake up a minute late without your favorite tune.
Furthermore, "Alexa, stop" worked to shut off the alarm even without a Wi-Fi connection, but phrases like "Alexa, snooze" or "Alexa, cancel" won't work. (Your mileage may vary on this point, one of my coworkers tested this with his Echo and while the alarm worked, saying "Alexa, stop" didn't.)
Amazon's support page even says that "if your device is muted or not connected to Wi-Fi, your timers or alarms still go off."
In the end, you probably don't need to worry about a possible Wi-Fi outage causing you to sleep through an alarm that failed to go off on your Echo---things like alarms and timers are all done locally on the device itself and don't need an internet connection to function properly.