I don’t think first contact with an alien technological civilisation has many parallels to contact among human civilisations on Earth. Any aliens we discover through radio or optical messages, either broadcast intentionally or inadvertently (for example, aliens picking up Earth’s missile attack surveillance radars whose signals radiate out into space) are likely to be hundreds or thousands of light years distant, so communication is effectively one way. So, it’s a question of what we can learn from the signals, if we are able to decode them. Since it is also likely any contact will be with a civilisation much older than ours (humans have only been able to broadcast signals detectable at interstellar ranges for about 75 years) they may have learned things which are useful, from basic science to technologies to ways to organise society that don’t involve mass killing from time to time.
I don’t think it’s likely that discovering a more advanced civilisation would demoralise humans and make them give up. A more typical human reaction would be to treat it as a “Sputnik moment” and, encouraged by the demonstration it was possible, start working on catching up. It would also provide an existence proof that success is possible, and that we’re not doomed to blow each other up or go extinct for one reason or another.
I don’t know the motivations of everybody funding SETI (in the U.S. every project is privately-funded: the U.S. science agencies do not support such work), but I believe a major reason is pure curiosity. The question “Are we alone?” is one which has profound implications regardless of whether the answer is yes or no.
If Robin Hanson’s “grabby aliens” theory (developed to explain why human technological civilisation developed so early in the habitable life of the universe), it is only curious civilisations that expand and change their environment who survive in the long term. A potentially grabby civilisation will be interested in possible neighbours and their history and capabilities.