Why can't I propose a non-simultaneous / non-reciprocal / hospitality exchange? The option is greyed out.

When you go to propose an exchange, only the exchange types that are accepted by the home listing are offered to you. If the home does not accept non-reciprocal exchanges, that type will be greyed out and you won't be able to select it. Often, people don't realise their listings are only set up to accept simultaneous exchanges by default so, if you've been discussing a particular type of exchange and find it's not offered, just send them a quick message to ask.