You're on the right track! Start adding your flour in the bowl as you did and when you can remove it without it being too tacky then start working it on a cutting board or countertop as you continue adding more flour. Sounds like maybe you pulled it out of the bowl too soon.
As for a specific ratio of flour to potatoes/veg. that's going to depend not only on the items you're using, but also cooking method (for instance I recommend baking potatoes for making gnocchi rather than boiling as they will be drier and need less flour added, thus making for a lighter dumpling).
Once you have enough flour in the mix so that you can knead it and work it, I usually do a "stretch" or "tug" test but holding the dough up in the air and with my hands at each end and then kind of pull and push the dough from the ends (hard to explain in text but imagine the push and pull of two magnets you're trying to push together). If it has a bit of tug and a little elasticity you should be fine. If it doesn't, then either knead it a bit more, adding flour only if it is sticky. If it still doesn't have any tug after kneading more, then you probably don't have enough flour. The gluten in the flour is what's going to help hold it all together so that you don't just end up with disintegrating mashed potato/vegetable blobs in your poaching water. If you're making a big batch it will probably take longer than you expect to get it to the right consistency.
As for types of flour, the more protein a flour contains the more liquid it can absorb. While I have not made gnocchi with bread flour, it would take less bread flour than all-purpose to absorb the moisture in the mix. It will also have a more distinct flavor from the add'l protein. One caveat is that it could end up making them too tough due to the extra gluten.
This is the last write-up on the topic, and it's much simpler:
For hand-pulled noodles, you need:
Bread flour (wet gluten 29-30%, protein 11%-12%)
45% added water
1% alkaline solution
kansui powder or (Lye Water + Baking powder) or Peng or Baking Soda
Ingredients (Alkaline solutions)
kansui powder
55% sodium carbonate (Na2CO3),
35% potassium carbonate (K2CO3),
10% sodium biphosphate dodecahydrate (NaHPO3.12H2O)
Lye Water
potassium carbonate (K2CO3) 74.5% in 100ml
sodium biphosphate (NaHPO3) 3.4% in 100ml
Instant ash Peng
sodium carbonate (Na2CO3) 90%
Sodium chloride (NaCl) 1.15%
It is recommended to use bread flour because of its higher wet gluten content compared to normal white flour. You will also need a small amount of alkali. One recommendation is to use kansui powder, due to its ph of 11. If kansui isn't available, the next option would be baking powder and lye water. If either isn't available, the next option would be to use sodium carbonate 1% of total weight flour weight (e.g. if you use 100g of flour, use 1g of sodium carbonate). If you're using a lower gluten dough like all-purpose flour, then baking powder will do just fine. If you're using cake flour, use baking soda.
(Remember to add the kansui/baking soda/baking powder/sodium carbonate to the flour and mix it in before adding water.)
Sodium carbonate vs. potassium carbonate:
I've haven't had success with bread flour, plain flour or all-purpose flour with potassium carbonate. Even though it increases the ph of the dough, it still does not give the dough its stretchability.
Why use alkaline solution in the first place?!
Alkaline solution will increase the water absorption and gluten creation of the dough to enable it to be pulled sooner than normal by leaving the dough to rest at room temperature for 45 to 60 minutes. The downside of the alkaline solution is that it also breaks down gluten after the peak absorption is achieved. So on one side you have faster absorption but on the other side the dough will become more resilient to pulling if left too long.
For a non-alkaline solution, use this tutorial. It will cover all the basics that you will need to learn, and you WILL need to use all-purpose flour or plain flour. Noodle flour works best.
In conclusion:
Without sodium carbonate it's possible to have hand-pulled noodles with any flour. The downside is you will have to knead the dough for 45 minutes and leave to rest for 2 hours until you can start pulling your noodle strands.
Without sodium carbonate the noodles are harder to pull and I had most success with cutting each strand and going from there from the video.
Best Answer
Hand pulled noodles are normally made with low to medium gluten flour (cake flour)
Alkalines, like Lye water or baking soda are added to soften hard flours (high gluten). You use much less for soft flour
Elastic may not be the word you want, supple is what you want in the dough. It need to be able to pull and not break, and stay that way
The dough needs to sit a few hours to fully saturate. Also keep it warm at a high-ish room temperature to keep it supple
It can take years of practice to pull noodles well