I don't get group texts with cellular data off I have an iPhone 6 running iOS 12.4.1.

I have a paltry data plan ($$$), so I've turned off iMessage in Settings -> Messages.

iMessages show up in blue in Messages app; regular text messages show up in green.

So far, so good. I can still send/receive messages when I'm out and about, and I'm not leaving data turned on and thus chewing through it quickly (and have it available if I need it, such as for driving directions).

Now, here comes the part I don't understand. Although I'm connected to my home network almost the time, much to my surprise, when I do turn on data, I get a number of queued messages. They all seem to be group messages, and show up in grey in Messages.app. [These groups may have been created before I switched off iMessage, if it makes a difference]

This is unbelievably frustrating. I had no idea I was missing messages. I don't get it -- why don't they come through as normal text messages, or come through over my WiFi? Why are they only available when I turn on ridiculously expensive cellular data?

More to the point, is there a way I can get them to come over WiFi, so I don't miss them and don't pay extra for them?
A one-to-one text message uses SMS. A group message uses MMS and this requires data. You would likely have the same issue if someone texts you a picture directly as this also uses MMS.

Check with your service provider though, as some include picture messaging (i.e. MMS) for "free" and don't count it as data. In which case you could leave iMessage enabled for data. YMMV.