The over-riding limitation with text messaging is there's a reliance on SMS and MMS as the only two protocols that all the texting services will support. Apple uses its iMessage protocol as its default, WhatsApp uses its WhatsApp protocol as its default, and just recently Google has been pushing its RCS as a new Android default. iMessage and WhatsApp are closed-source and proprietary, only in play when it's a point-to-point interaction (i.e. Apple user to another Apple user). RCS is not Open Source but Google has licensed it so it can be readily used by any texting service. But Apple has refused to adopt RCS into its iMessage app so at this point RCS is just another texting protocol. SMS and MMS are antiquated and have inherent limitations keeping up with today's technology, but they remain as the tenuous link we all rely upon as far as text messaging.
Your issue appears to be tied to how MMS is transferred. SMS takes up a minimal amount of bandwidth, and it can travel via WiFi, cellular, or Bluetooth connections. MMS messages can involve more bandwidth though, still a relatively low amount, but still more. Within common carriers (both sender and receiver using the same service), you 'might' be successful sending a MMS message over WiFi or cellular but in most cases you do need to use mobile data. Also keep in mind that any iMessage group text (and most Android group texts) are sent as MMS. There's an added issue where iMessage to iMessage and WhatsApp to WhatsApp interactions all take place within either Apple's servers or WhatsApp servers respectively, and that takes place no matter which carrier is in use. A benefit is since each of those closed systems transfer within each system, a user doesn't have to worry so much about file attachment sizes -- i.e. since all iMessage transfers take place within Apple's online servers file attachment limits are quite ample, but with MMS carriers will automatically scale down higher res photos and videos on-the-fly. It all boils down to all these variables are factor and conditional in different ways as to what device you're using and what device your recipient is using.
Text messaging is plagued by corporate oversight, there are multiple players involved and each is opting to use its own standard. Unlike email where that eventually became a common standard (POP and IMAP), by all indications texting doesn't appear to be heading in that same direction. So yeah, with MMS you'll probably need to use mobile data connectivity.