Quantifying DNA in a band on gel electrophoresis
All three methods could be used to measure the amount of DNA. However in practice, method 2 (estimation by dye brightness) typically works best in a normal workflow. It really depends on what you plan to do as your downstream application.

Problems with method 1:


Yield from gel purification methods is sometimes finicky and prone to loss.
Absorbance of remaining agarose can influence the results. DNA cleanup is a must.


Problems with method 2:


Quantification is difficult as you must estimate the band brightness against a known standard (add a known quantity of your ladder and use that for estimation).
Supercoiling of plasmids causes multiple bands to appear, so the estimation must be performed multiple times and then added (now it's less of an estimation and more a guess)


Problems with method 3:


Working with radioactivity adds a significant amount of safety protocols into a simple procedure
All products from radiolabeling must also be handled as radioactive material
Using an image has the same issue as method 2 where you're estimating instead of actually quantifying.
You could get more quantitative results from a scintillation counter, but then the products become more difficult to use because of purification from the scintillation fluid (maybe this is possible but I never tried it).
You're limited to procedures which would incorporate the radiolabeled nucleotides
If you do something like dA-tailing, then you're not quantifying the amount of DNA, but the amount of copies of a transcript.