I've never been a fan of using numbers in puzzles which don't intrinsically use numbers when solving, especially if they are really line/path puzzles like numberlink/fourwinds/anglers. I'd prefer to use notification to indicate how the path goes. Either S(traight)/T(urn) or H(orizontal)/V(ertical)/T(urn). Possibly for for example Four Winds, you could can use an End indication where to show the line ends in that square and for anglers a Fish indication.
I always find it annoying that I have to figure out the number of a certain path after i'm done with the puzzle as I would never put it in while solving. This will take more time the bigger the puzzle is.
1. My annoyance with such a method is probably similar to your annoyance with the numbers. Personally having a whole bunch of consecutive H/V/T makes it harder for me to check because my head is stringing them all together. It's a lot easier for me to verify it to myself if I can partition the row/column into discrete blocks corresponding to each number. For C9 in the example, there's a 3, then a block of five spaces for 11 (1), then a block of three spaces for 6, then another 11. On the other hand, if going through checking a test and seeing "HTVVVTVVTH" I wouldn't feel comfortable unless I went through the entire row/column space by space again counting. (also is that 7th space a V?)
2. The information content is far less and it starts getting into guessable territory. The constraints uniqueness places on a Numberlink solution means that for most rows and columns you could probably narrow it down to few reasonable straight/turn patterns.
For really big Numberlinks (like a nikoli selection marathon size) it would indeed be a hassle to use the numbers. But I've never seen such a size used in a competition.
Throwing out a possible alternative even though I don't like it myself for the ease of counting errors: number of turns in the paths that link a few selected pairs of numbers. Would also work for Anglers too. For Four Winds, you might also do something like count the number of dead-ends in each row, or in just a few marked rows.