Yeah, the baby meats are gross. Table food meats mashed up is better than the baby food meats.
Both of my boys had their first baby food jar at around 6 months old. I fed cereals at 4 months and then baby food at 6 months. A little after 9 months, they both had all of thier teeth, so I gave them table food cut up in a very small bites.
I did the same way as you did to make sure there's no allergies. I kept track of the foods passed the allergy test and saved the list to added to the baby's book keepsake. My kids have no food allergies.
About the meats, don't be discourage if your babies won't eat meats. It took my oldest to finally like eating meats when he was almost 4 years old. Now, my baby doesn't like meats, and he throws them on the floor. The doctor told me to keep offering meats on their plate even though you know it will end up on the floor. There's other ways to make sure the babies get their proteins such as yogurt, string cheese, cheeses, cottage cheese, etc.