What are the arguments in favour of the 'beginning' at John 1:1 being the old beginning?

First, there are no (good) arguments against it. John 1:1 is radically different than the start of the other Gospels, so "consistency" claims already fall flat. The argument that v6 must not represent a jump ahead in time is dodgy at best, as v6 is clearly starting a new thought. The later date of John's authorship also suggests that John may have been writing in response to early attempts to deny Christ's divinity, which is firmly established in many places¹.

Second, such an interpretation makes v3 inconsistent with the rest of scripture, and also... odd. Jesus is the Creator of... what, exactly (according to an interpretation that "beginning" here is something other than The Beginning)?

Compare:

Note that the greater context of Colossians 1 makes it clear that Paul is referring to Christ.

Thus, we have several passages, even in the Old Testament (when we consider John showing us that Christ is the Word; 1:1 and 1:14) that attest to all of Creation being accomplished through Christ. Why, then, would we think that John 1:3 is not meant to be consistent with these other passages? (Moreover, the passages in the Epistles that Christ specifically deny any notion that "the Word" is not Christ. Christ — the same Christ who became man, was crucified, and was raised — preexisted Creation, which is only possible if the man Christ is both man and God.)

There are additional passages that speak to Christ's preexistence; for example:

While these don't necessarily argue against the "beginning" in John 1 being later, they certainly support that a reading of John 1 as referring to The Beginning is valid. In fact, this is true even if one denies that the Logos of John 1:1 is the person of Christ, because numerous Scriptures speak of some sort of preexistence of something, and any alternate interpretation of those passages would also work with John 1:1.

Now, one argument is that John 1:1 and 1 John 1:1 are inconsistent. But as in the preceding paragraph, if we take John 1 as indicating that Christ has existed since the Beginning of Creation, then there is no trouble at all applying this same meaning to 1 John 1.

Perhaps most compelling, however, is when we consider how John's original audience would have read his Gospel. That original audience wouldn't be reading out of a collected volume where John is sandwiched between Luke and Acts. They would have been reading John's Gospel by itself. In that context, the most natural reading is either that vv1-4 are referring to an absolute beginning, giving context to a narrative that does indeed skip ahead at v5, or that v1 is a deliberate reference to Genesis 1:1. Of course, both possibilities result in the same interpretation. There's simply no context to imply otherwise, which is not the case in Mark (1:1: "The beginning of the gospel of Jesus Christ) or Luke (1:2's "those who from the beginning were eyewitnesses" correlates with the earlier part of the sentence in 1:1, "the things that have been accomplished among us"). Frankly, an interpretation as other than The Beginning is eisegetical; it's reading something into the text that simply isn't there.

Therefore, I conclude that any interpretation of "beginning" in John 1:1 as other than The Beginning is of no notable benefit, and is detrimental to theological consistency. Since there is no other overt reason to prefer one reading over the other, I am far inclined to side with the reading which is most theologically beneficial and is that accepted by the majority of Christians.

¹ Yes, yes, not everyone agrees with this assertion. Personally, however, I've always found pro-Trinitarian arguments to be straight-forward and for the most part "obvious" and refutations of anti-Trinitarian arguments easily made. Whereas I find the case against Trinitarianism feels like it's having to work to distort the meaning of Scripture. Anyway, that's your disclaimer that this answer is written from a pro-Trinitarian perspective. Plenty of words have been written on the subject elsewhere that I don't need to rehash here.