Using "that" after "demonstrate" and "prove" is perfectly fine. Actually, I would have to make an effort to find another correct way to use those words (and I doubt that it exists).
If the demonstration is in the current paper, it sounds awkward (to me, at least) to use the past tense. I would actually use the present tense, because you speak / write about something actual. Things would be different if you would write about a paper written by e.g. Isaac Newton.
Regarding the tenses used, there might be one exception: the rule-book of the university, about how to write papers. Sometimes they go into crazy details about what is allowed and what is frowned upon.
I just understood (in my own head) that other words can be used, with the respective change in the meaning:
- The paper proves if / when / where (something will happen).
- The paper proves what (will happen).
- The paper proves who (will do something).
Now I assume that other words might still exist :)
Other examples, kindly provided by @JasonBassford, proving that that is not always needed:
(1) The paper demonstrated a fair degree of intelligence.
(2) My efforts proved successful.