https://wa.me/their-number-with-the-country-code
Paste this link(after putting their number) in a chat with yourself (https://wa.me/your-number-with-the-country-code) and just click the link.
It is a built in feature. People can generate wa.me URLs for their account in settings.
That is a domain owned by WhatsApp. It's not some unofficial service.
This would all be easily fixable. You could be able to start a chat by entering (or, more realistically, copy-pasting) a phone number. And instead of a phone number, you could be shown the other person's profile name (like in Telegram).
There isn't much that's wrong per se, but when compared to e.g. Telegram it's much worse.
Outside of SMS, I only use Skype, Slack and Whatsapp for instant messaging, and in terms of spam, SMS, which is bound to my number, gets the most spam, which is already quite rare.
To be fair Telegram, privacy and security concerns aside, is a pretty high bar. Surely Facebook has the resources to make an exceptional app, but apparently they don't think it'd be worth it for them. I really wish something like Matrix had clients as good as Telegram's!