I do not think the sense related to existence (existential) denoted by the phrase 'there is' etc. is based on 'there'. The presence of be creates the meaning of existence.
The Online Etymology Dictionary gives the etymology of be as:
Old English beon, beom, bion "be, exist, come to be, become, happen,"
from Proto-Germanic *biju- "I am, I will be." This "b-root" is from
PIE root *bheue- "to be, exist, grow," and in addition to the words in
English it yielded German present first and second person singular
(bin, bist, from Old High German bim "I am," bist "thou art"), Latin
perfective tenses of esse (fui "I was," etc.), Old Church Slavonic
byti "be," Greek phu- "become," Old Irish bi'u "I am," Lithuanian būti
"to be," Russian byt' "to be," etc.
Also, the 'there' in the phrase 'there is' etc. cannot be an adverb of place.
In the sentence, "there is (there exists) a positive number that, when multiplied by itself, yields 25", there has a nominal function - to be the subject of the main clause.
There are vacant rooms upstairs. Here, if 'there' were an adverb of place, the sentence would have been an example of redundant usage because of the other adverb of place, upstairs.