Using an RF adapter to improve LTE signal in the house
would this setup work? NO

To make something like that work you would need to connect an antenna (I do not think I need an antenna: is nonsense, some form of antenna is always needed) to the input of the amplifier. The antenna needs to be for the correct frequency band as well. If not, then for example a strong WiFi signal can disrupt the operation of the amplifier, that will prevent you from receiving the LTE signal that you want. The output of the amplifier needs to be directly connected using a cable to your phone. You cannot just "retransmit" the signal through the air as then the amplifier would also be receiving it's own output signal. That might result in oscillations.

Similar to WiFi, for properly relaying/amplifying an LTE signal a proper repeater is needed. A repeater is much more than just an amplifier and also much more expensive. Since it needs to connect to the LTE network your cellular provider has to allow it to connect.

Your best bet: ask your cellular provider if they have a solution or try a different phone with (hopefully) better LTE reception or use a different cellular provider, some use the 800 MHz band for LTE (at least in my country) and 800 MHz can penetrate buildings more easily.

This RF adapter is quite pointless for what you want, cellular and RF is not that simple, you really need to know what you're doing.

Oh and GSM and LTE are not the same thing, they're different networks. GSM is "2G", the old/first network we mainly used for voicecalling and SMS. LTE is "4G" and much newer and can also provide high speed internet access. Compare it to if GSM is like a train then LTE is like an airplane. Both are for travel but the airplane is much faster.