Testing your design with a proper 4-4 still does not make it work. If the input belts are not full, it does not balance. Thus it is not a proper balancer. Thinking about it, my conclusions are these:
(1. The splitting from 2blue to 3red on the left side can be ommited completely. Feeding 3red+1blue or 3blue into a proper 4-4 gives the exact same output from the 4-4.)
2. But in any case, on the output side of a proper 4-4 in the middle of your example, you are facing a 4-3 problem. So you are transforming the 3-3 problem, which has a solution with a loop, into a 4-3 problem, which has a solution, that also has a loop. So you don't get rid of the loop in this example. It just uses more space and more belts, no matter of what color.
Ok, so far for the example you have given.
Now some general thoughts on balancing. A brute force way implementation of 4-4 balancing could be, to split each of the 4 input belts into 4 belts (=16 belts), interweaving them and then joining them again in a manner, so that each output belt gets 1/4th of each input belt. But what a hughe intermingled mess that would be. Luckily for n-n problems where n is a power of 2, very compact solutions do exist. How is this possible? Looking at 4-4, 8-8 and 16-16 balancing solutions, their commonality is noticable. They mix the contents of their belts in steps and in groups. And each group has a size that is a power of 2. 2 belts are mixed, then this group of size 2 is mixed with another group of size 2, then this group of size 4 is mixed with another group of size 4, then this group of size 8 is mixed with ... etc ... . This the basic concept of balancing and there is no other in vanila Factorio that I know of, except for brute forcing. Why? Because the basic element for splitting, which is necessary for mxing the contents of belts, is Factorios 2-2 splitter. It is the only element that can be used for mixing and only has 2 input and 2 output sockets. Thus balancing can only work for numbers of belts which are powers of 2. If Factorios splitter would be 3-3, it would only work for powers of 3.
Questions and answers:
Q: What about all the other n-m balancers, where n, m or both are not a power of 2?
A: If they balance properly, then they were 4-4, 8-8, ... balancers once, got excess outputs looped back to the input, got cut down and got their elements rearranged to make them more compact.
Q: Does brute forcing an n-m problem work?
A: Yes, but only if the number of output belts is a power of 2, as contents of each input belt have to by divided evenly by the number of output belts. With Factorios 2-2 splitter and without loopbacks, that is only possible for powers of 2. But brute forcing is a hughe intermingelation, happily traded for loopbacks.
Q: What about using ratio splitting to tackle the problem?
A: Ratio splitting without loopbacks also only works with powers of 2. 0.5, 0.25, etc. are powers of 2. It is used in
n-n (n=po2) balancers all over the place.
Q: What about using different belt colors for ratio splitting? Isn't it possible to split a blue belt into a red and a yellow one and then have 2/3rds and 1/3rd as output?
A: Splitting a blue belt into one red and one yellow is not proper ratio splitting. It only works if the blue belt is full and then it works because the items are backlogging at the exit to the yellow belt and are put on the red belt instead. The blue splitter always tries to put 50% of its inputs on each of its exits. If the input is 10 items/s the red and the yellow belt get 5 items/s equally.
Conclusion:
If my knowledge is correct and I did not make any mistakes in my thought process, my conclusion is: The concept to split to get powers of 2 and use n-n (n=po2) balancers to get rid of loopbacks is fundamentally flawed.
Edit: I don't mean this in any way negative. I want to drill down to the core of the concept presented here.
