A. Natural 100% pure honey today comes very clean without additives,and therefore there is really no halachic concern with it.
Honey with flavoring require supervision.
B. At the time of the gemara the honey sometimes contained bees or bee body parts like legs.
Naturally people removed the bees prior to eating but many times some legs were left behind which were hard to remove manually and people would eat the honey with the legs.
The rishonim asked on that custom how is it possible that people eat it and answered that legs aren’t forbidden when they are in the honey since they have no taste and even if the legs are cooked together with the honey it only spoils it which the Torah allows. Additionally its not like flesh that’s forbidden rather its like a dry bone (תוספות ע”ז ד”ה ההוא,ר”ן,רא”ש,מרדכי ועוד).
C. The Bet Yosef brought all those opinions lehalacha at the end of siman פ”א and in his shulchan aruch (see באר הגולה and the vilna gaon who point this tosfot to be the source of this psak).
D. This is also the psak of the rema in siman ק”ג,ב and even though he is talking about the utensil, see בדי השלחן that the rema really permits the legs as well. This is also the Chatam Sofer’s (הגהות על שו”ע) understanding that the rema agrees that it is permitted.
E. The achronim also allowed it see the pri chadash (פ”א,כז)who wrote openly that the legs are permitted to eat with the honey and he added that one needs to filter the honey only to remove the actual body of the bee and ants that might be there but legs that were left are fine.(some poskim by mistake misunderstood the Pri Chadash to mean that you need to filter out the legs but when you look inside the sefer its clearly not so).
F. Almost across the board the poskim allowed it based on all the above.
See the hashgachot like the star k and the ou that permit honey without supervision.
See also the psak of Rav Ben Tzion Mutsafi shlita.