Does it? Last time I used WhatsApp I could not use it on my desktop without scanning a QR code each time and keeping the phone nearby.

You need to scan the QR code only the first time using the desktop app.

Can you use the desktop app without the phone present? For example, if the phone is turned off.

I have heard you can, for about 2 weeks. Then the phone must be at least become active.

With Telegram I do not have to worry about losing my "primary" device in order to access my account. Telegram is more a social network than a chat app. Moreover Telegram does not require me to feed it my whole Contact List in order to communicate. WhatsApp has this limitation on purpose (there actually are workarounds but they are "hacks" and not how Meta wants you to use the app). It is very suspicious: why can't I search for a WhatsApp contact by manually typing in a number into WhatsApp? Instead I need to put it inside my Contact App and grant WhatsApp full access to it. LOL

There recently was a link making rounds here that purportedly allowed you to send WhatsApps to phone numbers not in your address book

Yes you can