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Exactly as thorough as you would have to be using any other kinds of symbols. It's just that vast messes of symbols are hellish for humans to read, but sentences aren't. Adding symbols to something doesn't make it more rigorous, less likely to be wrong, or really anything else. Symbols are useful for abbreviating in situations where this adds clarity, and making complex arguments easier to follow, but shouldn't be used where they do not help in this regard.

Yes they can and I'm of the opinion that symbolism and notation should be avoided unless it serves to simply the presentation of the material or to perform calculations. For example you want to cut a cube so that each face has a three by three grid of smaller cubes similar to the Rubix cube and with a little thought and experimentation once might conjecture that six is the minimal number of cuts. The best proof of this that I know of is simply "Consider the faces of the center cube." They require six cuts because there are six faces and it follows immediately. No symbols or calculation but still logical and mathematically sound.

Two points:

(i) Historically, all proofs were done in words—the use of standardised symbols is a surprisingly recent development. This is obscured a bit because a modern edition of, say, Euclid's Elements is likely to have had the words translated into modern notation.

(ii) Before symbols can be used they have to be defined, and ultimately that definition will be in words. It's easy to forget this, especially with ones that we use all the time and learnt in childhood. But, for example, we once had to learn that $2+3=5$ was short for "Two things together with three things is the same as five things".

Though a lot of us learnt instead that $2+3=5$ meant "Three things added to two things makes five things".

Now, these two definitions are different. One makes $2+3$ into an operation done to $2$, and treats $=$ as an instruction to carry it out; the other says that the number on the right has the same value as the expression on the left. The notation, though, doesn't make this distinction, and it's possible to spend years using the $=$ sign as though it meant "put the result of the operation on the left on the right".

So in this case we've got one string of symbols ($2+3=5$) a correct definition and a misleading definition. And how do we clarify the correct meaning of the symbols? By choosing which verbal definition to use. The precision is in the words (at least if they're well chosen).

Of course, more advanced symbols will most likely have some mathematical symbols in their definitions—but ultimately, we'll get back to words.

Natural language for expressing mathematical statements can be indeed vague and ambigous. However, when you study mathematics, one thing you will usually learn at the beginning is how to use mathematical terminology in a rigid, unambigous way (at least for communication with other people trained in mathematical terminology). This process takes usually some time if you are not a genius (I guess it took me about two years at the university until I became reasonable fluent), so unfortunately I fear I cannot tell you a small set of rules which kind of language is "right" for mathematical proofs, and which is "wrong".

Hence, the answer is clearly "yes, words are fine, when used correctly by a trained expert". Amazingly, one could say the same about more formal proofs using symbols.

Note that historically, before the 18th century, proofs using natural language was the de facto standard in mathematics. The symbolics we often use today was developed in the 18th and 19th century.

Behind the proof system is logic... you need to write a reasoning that is fool proof and can be reproduced by the reader to lead to the same conclusion, and every step of the proof must be unambiguous and without "exceptions" (if there are special cases, they must be stated). As long as this is respected, the proof is correct and complete. When you see a symbolic proof, you can still read it in plain language, as long as you understand what it means, so there is no real difference (as long as the proof is rigorous, without "holes" or ambiguous statements).

Note that this excludes statements such as "this is obvious". You need to tell the reader of the proof what steps to take in his own mind to come to a single unmistaken conclusion. This part is very important - not understanding this leads some people to rejects proofs as opinions (all pseudoscience relies on this fallacy).

Now, just as words are just notation for thoughts, so are symbolic expressions just short notation for longer words. Symbolic notation has the advantage to being language-independent, and exact within their previous agreed upon definition. They a lot of times simplify things in algebra, arithmetics and functional analysis, where reasoning just follows simple steps without decision making and reasoning.

However, when it comes to logic, deduction, and other high-level thought processes, notation gets clumsier and a lot of times harder to understand (there are symbols for "therefore" and statements such as "A implies Β", but the author might not choose to use them). Instead of calculations, you have something that very much resembles formal computer programs, and fewer people are trained to read them fluently.

Think of lawyers: law is written in "english", but most "everyday english" isn't used, because it's ambiguous. Instead, the words are meticulously put together to try to cover all the corner cases and have only one interpretation (so much, that for a layman, the text is almost incomprehensible). The metaphor is not the best, because in lawmaking, there is no rigorous foundation (no true axioms) to rely upon, but I hope you understand the point.

For your particular example:

Just keep distributing $A$ over and over ad nauseum and you get the term on the right.

would not be a convincing proof. This is not because it is in words, however -- words are perfectly fine.

But it fails to convince because the intersection is over an infinite family of sets. Your proposal would work fine for a finite intersection, in that it gives a recipe for constructing an algebraic proof that would itself be convincing. And in ordinary mathematics a convincing recipe for a convincing proof is itself as good as the real thing.

But for an infinite intersection, the algebraic calculation you're describing never ends! No matter how many steps you do, there will still be an intersection of infinitely many $A_i$s that have yet to be distributed over in your expression. So your recipe does not lead to a finite proof, and infinite things (to the extent they are "things" at all) are not convincing arguments.

There are ways to convert some cases of infinitary intuition into actual convincing proofs, but they have subtle pitfalls, so you can't get away with using them -- no matter whether with words or with symbols -- unless you also convince the reader/listener that you know what these pitfalls are and have a working strategy for avoiding them. Typically this means you need to explicitly describe how you handle the step from "arbitrarily but finitely many" to "infinitely many" (or in more sophisticated phrasing: what do you do at a limit ordinal?).

A somewhat unheralded part of mathematics education is that over time you will get to see sufficiently many examples of this that you collect a toolbox of "usual tricks". When communicating in a situation where you trust everyone knows the usual tricks you can often get away with not even specifying which trick you're using, if everybody present is experienced enough to see quickly that there's one of the usual tricks that will obviously work.