I'm an amateur looking for help! I talk about "heads" and "tails". Other examples of tails are "don't you think?" "do you see what I mean", "or something similar". They are phrases L2 beginners can use to invite corrections or find out whether they have been understood.
Many branches of linguistics deal with these types of clauses but I don't recall ever seeing them classed under one term. This is because, they are structurally and functionally quite diverse.
Some of them come under the category of hedges (or something like that), others conversation repair or maintanance (you see what I mean), or turn taking prompts (don't you think), others still might be question tags (wouldn't you), etc.
They're syntactically quite interesting because they often look they're a part of a sentence or a clause but discursively do something different. They would also often follow different intonation patterns.
However, despite not being a very high profile part of linguistic investigations (undeservedly), I'd say they are quite crucial to L2 learners seeking to develop their fluency.
I like your term 'sentence tails' but you're probably better off calling them something like 'conversation boosters'. Especially since the term 'head' is already used in linguistics for something else. Hopefully, somebody will contribute some better terminology (and it's also possible, I'm overlooking something quite well known).