Identify which direction is missing
There are two separate paths: EveryBand reading records already saved in Apple Health, and EveryBand writing permitted records to Health. A successful import does not prove export is enabled. A record visible inside Boop is not automatically eligible for a Health write.
Start with one missing category and one time period. For example, check last night's sleep rather than describing the entire connection as broken. The installed app remains named Boop while the website uses EveryBand; look for Boop when reviewing its access.
Check TestFlight for the build available to your device. Setup labels and supported behavior can differ between betas.