Can mobile phones use email instead of SMS, without traffic?

No client is needed except SMS. You can send it as long as you have a number. Everything else needs traffic. It is wrong to rely on signals. If there is no signal, it is turned off or in flight mode, or you don't insert the card. In that case, many functions of the mobile phone will not be opened, let alone used normally. You can try the walkie-talkie, contact frequently in short distance, and there is no telephone charge.