When you read that で is the 連用形 of 断定の助動詞「だ」, that's a reanalysis. That is, from a modern perspective it makes sense to talk about it that way, and you should be familiar with the concept if for no other reason than to understand dictionary entries.
But that doesn't mean that's how it came about historically. Of course だ is contracted from である, and etymologically で is just である without the tense-bearing ある. The same thing is true of the old copula なり, which is a contraction of にあり. If you look it up, you'll see that it's analyzed as having に as a 連用形, but from an etymological perspective it's simply にあり without the あり.
To assert that it's etymologically a だ plus て, we'd need to suggest something like にてありて somehow contracting all the way down to で, but this is implausible historically, given that に and にて were used as copulas before the derived copula である existed. The copula に goes back all the way to proto-Japonic and is clearly a more basic form.
You also ask "is it still correct to isolate the ある and say it means 'to exist'." Here I have to say no.
It's clear that in Modern Japanese ある in である is being used grammatically, not lexically. "Exist" is a gross over-translation. If you'd like to suggest that this was where it comes from etymologically, that it's its literal meaning, you could do so (although I think the "as" meaning of で is historically secondary, so I don't think it's quite right). But in any case I don't think that's what it means now.
Remember that etymology does not determine what a word means in the modern language. To determine that, you have to look at how it's used now. Otherwise, you're falling victim to the etymological fallacy.
So how do we analyze it now? Well, there's more than one valid way to think about it, and I won't attempt to teach you the One True Analysis. But I'll present one possible alternative below.
The modern linguist Kunio Nishiyama, in his 1999 Adjectives and the Copulas in Japanese, refers to で as the predicative copula and ある as the dummy copula. In this analysis, the former is used to create copular predicates, and the latter is a dummy verb whose sole purpose is to bear tense. (Of course, ある has other uses where it is not a dummy verb.)
If we adopt this theory, we can treat the combination of である contracting into だ when the two are next to each other. This contraction only happens under certain conditions. For example, if you insert a particle such as も, the contraction is blocked:
Ａで Ｂで ある → Ａで Ｂだ (である becomes だ)
Ａでも Ｂでも ある → ＡでもＢでもある (contraction blocked)
Here, both Ａで and Ｂで are copular predicates, but it's not a complete sentence without ある, which turns the whole thing into a present tense sentence.
In similar fashion, we can treat だった as contracted from であった and です as contracted from であります if we'd like (although the actual historical etymology of です is uncertain).
