For example, before Rome reached its heyday, was there any prevailing idea that they were actually on the decline? Did any Chinese dynasties know they were about to meet their end as they began to decline?
I am not entirely sure America does have an obsession with its own decline.
Edit: Further Context, due to the 20 year rule this is a poll from 2000 https://news.gallup.com/poll/2434/americans-overwhelmingly-happy-optimistic-about-future.aspx
Here you can see them talking about Americans being optimistic for the future. This suggests that Americans in 2000 had less of a belief in the future decline of America. The poll in 2000 had 78% reporting they were Optimistic or Very Optimistic in relation to this question "What is your outlook for the United States during the next 20 years? Would you say it is -- very optimistic, optimistic, pessimistic, or very pessimistic?" By comparison 1979 only had 51% reporting some form of Optimism for the future of the country.
If you're referring to how politicians like to rile up their base with talk of how our past is better, you may want to look into how Fascists, Populists, and other manipulative politicians evoke emotions involving nationalism, fears, and loss to get their way.
China, for instance, leans heavily on their Han Dynasty past. Italy leaned heavily on Rome, and Germany infamously on their mythical Aryanism.
Largely the process is to call attention to how things were better, and how someone strong and powerful is necessary to get back to that greatness.
As for Rome specifically, this older answer by /u/Capt_Blackadder addresses it: https://www.reddit.com/r/AskHistorians/comments/20vb7l/did_romans_realize_their_empire_was_in_decline/cg78w3s/
You'd have to be more specific about who is talking about decline, what kind of decline, and what you mean by obsession. There is a common tendency to talk about how things are in decline. The next generations are always less moral, less able, less great; and the older generations are always regarded as better. In general there is this trend to feel like things are always in some form of decline, even if that may not be the case.
Edit: To give some examples. Horace around 20BC in Rome says "Our sires' age was worse than our grandsires'. We, their sons, are more worthless than they; so in our turn we shall give the world a progeny yet more corrupt."
This post by /u/QVCatullus calls attention to Hesiod doomsaying how everything will get worse: https://www.reddit.com/r/AskHistorians/comments/32g1r2/what_was_the_first_written_example_of_the_well/cqb42w4/