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This sequence: 可以不用 sounds odd to me. Maybe for Chinese ears it sounds perfectly natural. I suppose my brain immediately puts it in English, but I think the meaning is not 'you can not use' but rather 'you don't need'.
Can I replace 可以不用 with 不需要 here?
如果想准确地知道自己到底消耗了多少卡路里，平时可以不用计步软件或运动手环，只在开始跑步的时候使用就可以了。
"可以不用" and "不需要" can be used interchangeably when it means 'don't need to' and followed by a verb or verb phrase
Example:
[可以不用][付钱] = [不需要][付钱] = [don't need to] [pay]
[可以不用][用糖] = [不需要][用糖] = [don't need to] [use sugar]
~
"可以不用" (able to/ allow to not use) followed by a noun and "不需要" (don't need) followed by a noun, have different meanings
Example:
[不需要][糖]= [don't need] [sugar]
[可以][不用][糖] = [able to/ allowed to][not use][sugar]
Example of "可以不用糖" in context:
"我可以不用糖做甜品嗎？" (Can I make dessert not using sugar?)
~
[(可以不用)(用糖)] = [(don't need to) (use sugar)].
It is grammatically correct, but "可以不用用糖" (don't need to use sugar) may sound a little bit awkward in Chinese (two 用 back to back). Better change it to 不需要用糖
"不需要 + verb" = "don't need to + verb"
"不需要 + noun" = "don't need + noun"
"可以不用 + verb" = "don't need to + verb"
"可以 + 不用 + noun" = "able to/allowed to + not use + noun"
Can I replace 可以不用 with 不需要？
Yes, but only when it means 'don't need to' and followed by a verb, not noun
平时可以不用计步软件或运动手环
计步软件 is a noun, therefore, "可以不用计步软件" means "able to/ allowed to not use steps counting app"
TL;DR You may often replace 可以不用 with 可以不需要 without changing the nuance, but not in this particular sentence because "用" describes the action of "use", which would be lost if the word is replaced.
The actual difference here is the addition of 可以 before 不用/不需要. Because 不用, 不需要 are synonyms. The addition of 可以 brings up a suggestion, another option to the listener.
The following are definitions provided by the Taiwanese Ministry of Education.
不需要 is listed as a definition for 不用.
However, in your particular example, 用 also provides the sense of "use" in the clause, which may be lacking if "用" is directly replaced by "需要".
In here, "可以不用" takes on its primal meaning: "don't have to use" (more literally: is allowed to not using) but "可以不需要" would just mean "don't need" (more lit.: is allowed to not needing). So in the end, you may have changed "You don't have to always use a step counter or sports band" into "You don't always need a step counter or sports band".
So we would need to reinsert "用" somewhere else, like: 不需要使用/不需要用 (no need to use), where 用/使用 would be grammatically analyzed as being appended to the object to form a so-called "verb phrase", or an object which is just another predicate.
However, using the common phrase "不用" elsewhere, without the action of "use" would be synonymous to "不需要".
So here, I will continue to analyze the connotation of adding "可以-".
I believe the difference completely lies in the intention of the speaker. The effects of the sentences are exactly the same, where the speaker suggests doing something without the object.
可以不用/需要 - may do without. The speaker suggests leaving out the object, but it would have been fine otherwise.
不用/需要 - no need. The speaker suggests leaving out the object. Adding it back anyway might not be appropriate. In my personal opinion as a native speaker, this may a more polite way of saying "should not".
I have the following examples with their situations.
I do have something relating to grammar that I do not agree with @TangHo. They suggested these phrases may be paired with "+ verb" or "+ noun". However, I believe in linguistics analysis, "可以不用" and "不需要" are predicators/verbs. So adding another separate verb afterwards may not be appropriate.
The same goes in English, there can only be one predicator in a clause. As such, the sentence segment that comes after should be the object/complement. So a separate verb cannot come after "可以不用" or "不需要".
In English, the action-showing word placed after the finite verb would be called a gerund, which is analyzed as a noun, acting as the object. In general, this is called linguistically, Conversion or in Chinese, 轉類. In Chinese, part-of-speech conversions may be invisible, but in English, the spelling often changes.
I've given examples of appending "objects" above. Here are examples using "complements": "There is no need to be that quick."
如果想准确地知道自己到底消耗了多少卡路里，平时可以不用计步软件或运动手环，只在开始跑步的时候使用就可以了。
According to the context, 可以不用 should be understood as 可以不使用, which means don't need to use. This is indicated by the following sentence "只在开始跑步的时候使用就可以了".
P. S. One of Chinese tricks is that 可以不使用 could sometimes also implies 可以使用. lol.
可以不用 is more like a recommendation than a judgement.
不需要 is more like a judgement than recommendation.
可以不用 is more polite than 不需要.
However, the difference is very tiny, almost neglectable.
