The iOS app gives you an option to manually punch in a number. I'm not sure whether that's because it's to comply with app store regulations, or the product owners on both sides prioritizing different things. On both platforms you can still initiate chats without contacts access by using the wa.me domain, eg. wa.me/12125551234 for +1 (212) 555 1234.

Sometimes I need to contact somebody on Whatsapp one time or such, and I don't want to save their contact. But Whatsapp on Android shows only my own contacts to initiate chat. Like you said, wa.me is a way. I made this quick page for exactly this:

http://spa.bydav.in/num2wa.html

Alternatively on Android, the app "Click to chat" is a tiny utility that directly opens up a WhatsApp chat with any given number:

https://play.google.com/store/apps/details?id=com.trianguloy...

The fact that this is necessary is wild.

That's not true for me on iOS. I denied it access to my contacts, and I can still start a new conversation by entering a recipient's phone number.