Tonight's dinner was a beef with rice wrapped in lettuce leaves. It was tasty, but very messy and time-consuming to wrap. Is there some trick to this that makes it easier?
Yes, they can. Most people I know who make them for the first time just use too much filling.
Traditional leaf-wrapped dishes (I know them under the Turkish word
sarma) use a softened leaf (use pickled ones, or blanch fresh leaves) and a small amount of filling. Use one heaped tablespoon of filling on a 10 cm leaf, then wrap tightly.
A demonstration with sauerkraut sarmi is available on YouTube: enter link description here. Watch 2:10 to 2:30.
The end result does not fall apart, even after cutting into it. (grape leaf on the left, sauerkraut on the right).
Not sure if this is what you're looking for, but rather than trying to solve the problem, you can also just accept it. Bring it all to the table, and just make wraps as you eat them. It won't make it less messy, but it won't be much slower than eating pre-made wraps, and you'll be spending that time at the table with people instead of waiting to eat.
To reduce the mess at least a bit, go for leafier lettuce, avoiding the parts with thick, stiff ribs, and definitely avoiding varieties like iceberg. (I've seen people recommend iceberg for wraps, but it seems like an awful choice to me.)
You can also potentially use other greens. There's napa cabbage and blanched cabbage, but also things with a bit more flavor that you might find in an Asian grocery like perilla (shiso, kaenip, ...) - I'm not sure I've had all of them, but the leaves used for ssam are probably good ideas.
Blanch the leaves briefly (I like cabbage over lettuce)
Instead of minced (ground) meat use a decent sausage (whole meat). Prick the sausage skin well so it releases it's cooking juices into the cabbage
Add sauce inside or outside the leaves (as you prefer), and bake as you like
Quick and simple, little mess, tastes almost the same