The order of berachot before eating foods are as follows (Mishna Berurah 211:35): HaMotzei, Mezonot, Hagafen, HaEtz, HaAdama, and Shehakol.
I seems as though the more חשוב the food is, the greater precedence it has and the more significant the after-beracha is.
Meat is considered a delicacy, a facilitator of (non-subjective) simcha (according to the Rambam, see here), is used in important korbanot, has many halachot regarding its preparation, has a higher level of 'soul' in comparison to inanimate vegetables (Ramchal), etc.
Quote from the Rambam (Hilchot Yom Tov 6:18), talking about simchat Yom Tov:
והאנשים, אוכלין בשר ושותין יין, שאין שמחה אלא בבשר, ואין שמחה אלא ביין.
Why then does meat not have a greater significance in terms of berachot?