In order to get some early beetroot - but the result has been lots of leggy beetroot
I don't know what area you are in but even so I think you have been a tad early in sowing them. It will be another 4-5 weeks before I sow mine using the same method as you, then I sow another batch directly into the soil at the end of May.
Your problem has been mainly 'lack of daylight' a common factor with early sown stuff, as you have found out it is relatively easy to germinate stuff but growing on can often be a problem due to lack of good light, and day length.
Rather than throw them out you could plant a few out and cover them with a fleece or glass cloche.
I would guess your ground will be quite cold so to alleviate that you can form a drill about a couple of inches deep and place some new multi-purpose or potting compost into and transplant your seedlings into it. At least then the soil will be warmer and dryer and the cloche might give enough protection for them to develop.
Other than that start again in another month or so.
Sorry! but that is the best advice I can offer!....Tg