Long ago, arithmetic used to be taught at university level. Then people upgraded their thinking from tallies such as IIII to symbols such as 4.
From being 'university' trained abacists, people were then able to become algorists simply by mastering their times tables! Hindu Arabic Numerals and base ten had an inbuilt logic engine!
I do not see a problem with grade 4 kids being exposed gently to algebra as letter games (numlets) in which they need to solve puzzles. This could be an 'extra' for fun or as a project seperate to a formal curriculum.
What I object to in your PDF is the use of terminology that is way too obscure/advanced for kids! The PDF you shared looks like it was written by a mathematician and not an educator.
Apart from having tutored kids, I have enjoyed raising three children. So Bob we both have direct instruction experience with our own children. Whenever I would introduce an idea I would always do it in a way that was (relatively) fun.
In Australia children have letters attached to their year grade based on their teacher's surname. So a child may be in grade 1M with Ms Miller or in grade 1T with Mr Tendulkar.
As Aussie kids already know number/letter combinations in this way, my experience is it's better to work with letters as labels long before they represent variables.
This can then move onto ideas of sets or even be taught alongside the idea of sets.
Remember how kids learned A is for Apple through to Z is for Zebra?
Six apples, three bananas and two carrots are put into a grocery bag. You can no longer see the items, yet they are still there!
So we can draw a bag and write the 'numlets' 6A and 3B and 2C on the bag.
Take out two apples and you now draw the bag with 4A and 3B and 2C on it.
The letter N is for Number so 2B + 5B = NB what is the number of Bananas? Seven! N = 7 so 2B + 5B = 7B.
Then with $2.00 you can buy one banana or two carrots!
So the numlet cost and swapping conversation becomes 1B = 2C.
Swapping food at lunchtime is what kids do at school, so this way of introducing object labels alongside numbers is easy. You can now swap letters, whoops I mean food.
Counting means you are adding the same thing. So you can count the number of fruit OR you can count the number of vegetables. You can also count the items of food that include both fruit AND vegetables. ie sets
Just as one-to-one correspondence is at the heart of math, for every A there is an Apple.
Pretty soon kids think nothing of combining letters with numbers because there is a concrete model behind the logic.
Provided you keep the pedagogy concrete rather than abstract (unitless) kids can devour the logic of algebra and find it quite yummy.
The problem is not the math, it is the pedagogy and the preference to have one-on-one interaction with the child.
Parents have that opportunity yet not the knowledge, while teachers (may) have the knowledge yet not the time with 20 kids to teach.
So in answer to your question, 'Why?' I say 'Why not?' provided the ideas are age appropriate.
My first ever experience of being excited by math was when my big sister (5 years older then me) taught me the Pythagorean theorem when we were meant to go to sleep in a tent. She only mentioned the area of squares. How much more fun would it have been had she also told me those same line segments on the triangle also produce triangles, circles and other shapes that have the same area relationships!
Sometimes math education can just be too slow! From liking math at primary school I later flunked it at high school. It got too abstract and meaningless and way too boring!
Keep math fun and challenging and like a wordfind puzzle or getting through the next level of a game, kids will usually want more!
I know this grocery bag example may not be 'serious algebra' yet I do think it's less efective to introduce algebra in grade 6 without having undertaken a simpler stage of number and letter combinations and puzzles.
Lastly, just in case you haven't seen it, I have attached a Singapore math book topics by year level chart designed for the US. It may be a little less advanced than the true situation in Singapore.