No you can sow beetroots pretty much until mid summer. I sow a few into modules and then plant the modules out in the spring [put my first lot in under a cloche last week], and once the first batch are in, I'll resow and do that again around 4-5 times throughout the summer. They can be left in the ground until they are needed.
I just dug some up that were sown last autumn, and left all winter. They are a very reliable crop.
Another not well known thing about beetroot is that if you leave them in the ground long enough, they will flower in the summer of their second year. The stems grow taller than we humans are [so need staking], and loads of tiny flowers will bloom on downward hanging stems, and the scent of those flowers is one of the best flower scents I've ever known.
If you leave 3 to go to seed, or even 2, you will get enough seeds to last probably the rest of your life, or to share with many, many other people, and they will still be viable if kept cool for around 15 years. So saving from 2-3 beetroots every decade is enough to keep you in beets forever.