I'm really sorry about everything below. It's very self-indulgent and not very polished or good.
I'm not the best person to ask about what the normal opinion is wrt the place-value notation. A lot of people obviously deal with it daily and live with it, but I'm sure some of them find it hard. I've always been fine with it though of course calculating with numbers is very bothersome. I am very happy to rarely have to see anything other than rational numbers and in them natural numbers larger than five.
However place-value notation is just a very convenient way to pack a lot of abstract info into simple notation and this is an idea that mathematicians rely on to do their work (see the second example below). Humans have very limited working memory. To be able to grasp complicated abstract things which contain too much information for this working memory you have to somehow be able to see the big picture that is composed by all of this small detail. The main tool in this is just building intuition about the things you're working with. This is basically "seeing" or somehow feeling the understanding you have of some mathematical object.
If you're doing some kind of geometrical argument in your head you're in some very general sense seeing your argument happen in your head. The geometrical object you're thinking of might be higher (than three) dimensional or might be too not-nice (somehow too abstractly defined) to have a definable dimension, but you're using your intuition of the dimensions you do know (one-, two- and three-dimensions) and the nice objects in them to help yourself understand the object. This of course has many pitfalls and you spend a lot of time being scared straight by your teachers with pathological examples of how bad and non-intuitive things can happen in the not-nice cases and why you can never assume anything. This never assuming anything and being scared of uncertain things is a fundamental skill in doing math.
(It has also probably worsened my mental state and made me more suicidal since I am now very bad at dealing with uncertainty. I might have just been bad at it before as well and that's why I gravitated to math, but idk.)
That is also why learning math is so slow. You're building from the simplest of definitions back up to the nice properties you know and learning under what assumptions they hold and when you can assume things to "be nice". The first example below is a proof from a book about 3-manifolds (nice three-dimensional geometric objects) and is incredibly easy to read (for me at least) (also notice how it is completely in natural language and has very few symbols). However for it to be a proper proof the writer and (hopefully) the reader has to have read huge amounts of math to know for sure that things actually do work as nicely as it says. Topology (the most loose form of geometry) starts with just a "set", a collection of objects which we call points, without any predetermined structure and a "topology", a collection of subcollections of this set, which in a (very abstract and convoluted) sense tell us which points are close to each other. You study these kinds of very abstract geometric objects enough to know the pitfalls and then you can using this understanding of topology define (in an imo not so convoluted way) a manifold, which is just a geometric object that locally looks like the nice geometric objects we're used to imagining.
As an example you can take the Jordan curve theorem which just says that if you have (a potentially pathologically) wiggly circle in two-dimensional space (or an (n-1)-dimensional sphere in n-space) that it actually divides that space into an inner and outer part. An at first intuitive fact that we just couldn't be sure of (and shouldn't have been sure of).
You can look at the second example which is from a more algebraic form of geometry. You have a lot of this referencing back to earlier results written out. You can go back a couple or ten or a hundred pages to look at the statement x.xx which is going to just be some kind of fact about the mathematical objects you're working with. When you read it you hopefully understood from the proof why that fact was true and you gained some greater intuition about the object. Therefore when you see that reference you might have to go look back at it, but it might be that you don't because you can guess from the context that oh yes, I can see where this argument is going and that this obviously follows from this fact in my intuition. Therefore it isn't necessarily that much harder to read than the first one.
I wanted to choose the second example to show something about notation, but now that I look at it, it doesn't really say anything. We just use a lot of symbols and well-chosen notation and they have a lot of inbuilt meaning for us. You can see that even though the proof isn't actually calculating anything using the symbols, it is saving a lot of space and a lot of working memory for us by using well-chosen notation.
There's probably lots of videos from places like numberphile etc. where they explain how a mathematician works in a way more concise and simple way. Sorry, I just kind of got carried away with writing this.
