Personal temperament seems to be a big factor in how people respond to these situations. I see a match against a better player as an exciting challenge, and I'm looking for ways to win. But that comes to me naturally.
One of the things I try to consider at the start of my innings is if I let the other guy back to the table (without a safe), given the layout, is he going to get out. Against a very strong player, the answer it more likely to be yes. (In which case I'll look to improve my layout, worsen his, and safe, at some point in my innings. Or try for a harder out than I otherwise would.)
I remember once playing against a (slightly) better player who didn't know my game well. We were about even when I broke and ran two straight racks (which put me ahead by one). He convinced himself he was outmatched and maybe won one more rack in the match.
That kind of temperament issue, I think can be overcome. If I'd pointed out that he was still about level and playing well when it happened, and that anyone capable of running racks will sometimes string a couple together by chance, maybe that could have changed his outlook.
I didn't tell him anything of the kind—why help your local opponents improve? But I kind of regret it now. Its only amateur pool, and you can help someone out, maybe give them a more positive outlook...
I'm also reminded of playing poker (live, small money) and keeping quiet to avoid giving things away. In hindsight, whatever, if anything, I'd give away was probably worth less than engaging with people, sharing ideas, etc.