UHF TV transmitting circuitry
Can I just hook up something like this https://www.ebay.com/p/533392191 just to transmit signal aerial?


No, It is an LNA (Low Noise Amplifier) and LNAs are not generally used in the transmitter side because of their low IIP3 values ( i.e. LNAs will easily distort input signal when the input power is quite large. They are excellent amplifiers when input power is low and thus they are used in receiving end). For transmitter side, you should use power amplifier. 


  Do I need to use a wideband amplifier because I also want to use a stereo sound (MTS/NICAM/A2)?


Sound type, mono or stereo, is not relevant for deciding bandwidth of RF amplifier. The amplifier you choose should cover the all TV channel frequencies you would like use. As you said that you are operating in UHF, so 0.1-2000 Mhz power amplifier would be a good choice.


  Wouldn't it be simpler to design RF modulator + amplifier from scratch to overcome aforementioned issues?


No, if you don't have a good electronics background. Even if you are experienced with designing such stuff from scratch, I am sure you can get readymade things which you can connect together and get your setup working.