On the rare occasions I've had to deal with Shimano triple front shifting over the years, I've been surprised at how dependent successful shifting is on having the cable at higher tension than I think it should need. There are probably a lot of how-to videos on getting it right, but one method might be this:

(1) With the chain on the small ring and the shifter clicked into the position where the cable is slackest, turn the barrel adjuster at the lever until it's a turn or two out from fully tightened into the lever.

(2) Shift from small to middle (as you've succeeded in doing before), click the shifter into its most relaxed position, so to speak, and then take a good bit of the slack out of the cable (using a pair of pliers if necessary) and snug up the pinch bolt.

(3) Shift to small, then middle, then large.

If the cable tension is still too low, the chain won't make it on the large, or won't stay there when you release the lever. If too high, the chain will go on the large but won't make it back to the middle. In either case, follow the steps listed above and try again.

Once it's working almost but not quite perfectly, use the barrel adjuster to get it just right.

That's all from memory. Others may come along with much better info. Or double-check with, e.g., a Park Tools or some other pertinent video.