I don't have an answer, but in case it helps I grow virtually all my veg in pots and plant out.
In the main I grow the plants such that they are potted-on to 9cm pots. These fit neatly into holes made with a long-handled bulb planter, so I can make all the holes, at least, standing up. No thinning required of course (I prick-out 3x Beetroot per 9cm pot and harvest the largest-first leaving the remaining ones for a later harvest, but everything else [I think/!!] is one-per-pot).
Parsnips are grown in "pots" made from rolling a sheet of newspaper around an aerosol can - they too fit neatly into holes made with a bulb planter.
Carrots I sow direct - too much trouble to sow individually in, e.g. Loo rolls, for the quantity I need. I sow them thin, and don't thin them thereafter, as such, but I harvest every-other-one when they are small-ish to provide more space for the others to grow on some more / bigger. Mixing the seed with sand, before sowing, can help if you are not good at sowing thinly.