On AT&T, there's LTE, 4G, and 3G, 4G and 3G are basically the same thing, but LTE is much faster than the two. 3G technology on AT&T is HSPA (High Speed Packet Access), and 4G is HSPA+. Most of AT&T's 3G service has been upgraded to 4G, but it's still technically 3G, and it can be very slow, so I can see your concern about not getting LTE.
As for your LTE not working, make sure to go into Settings-Cellular-Cellular Data Options-Enable LTE, and then enable it. I enabled mine on Voice and Data. If it's on Data Only, your phone will switch to 4G/3G when your sending a non iMessage text, placing a call, etc.
Try hard restarting your phone by holding the lock and home button until you see the Apple logo, and if that doesn't work, you can also try to reset your network settings by going to settings-general-reset-reset network settings.
It could also be based on the signal strength in your area, because in one area, I'll get 1 bar of LTE on my iPhone 6s, but then my phone will switch to about 3-4 bars of 4G because the signal is stronger. But in that same place, my friend with the iPhone 7 with the same carrier as me(AT&T) stays on LTE. Are the other people with the iPhone 6s in you family also having this issue? Does their phone switch to 4G often?
If non of that helps, it might be a hardware issue with the cellular modem. If you bring it in to Apple, they might be able to replace it for free.