Finishing PhD early and becoming my PI's postdoc instead

Regarding potential downsides for you:

"Time since obtaining the PhD" is often taken as the academic age, so by graduating early, you make yourself appear older. This can have a formal impact, for example, many grants meant for postdocs come with age restrictions, so are only available in the first n years after obtaining your PhD.

[Side note: This wouldn't matter for you if you get a faculty position straight away, but at least in my area, this would be quite unusual.]

The academic age can also play a role informally if people have to judge your productivity. (Number of papers)/(Years since PhD) is a formula I've seen a few times in this context.

You don't mention the country/field, but I'll give my 2 cents because I did my PhD in 3-4 years (depending how you count it).

IMHO and experience, there are virtually no upsides to rushing this:

• If you don't have a postdoc position guaranteed, you would be competing against people that spent 6 years padding their resumes. You have 3 papers, they will have about 10-12, or at least 5. The last years tend to be more productive.
• Postdocs usually have limited duration. Effectively, you would be reducing the amount of time you have to "pad" your resume for a position after the postdoc. Think of someone with 6 year phd in their belts + 2 year postdoc...
• Postdocs can teach as well. It is good for your CV and can help financially.
• Postdocs are shorter. It always takes a while to get your bearings at the start, so the scientific production will likely take a hit.

Of course, you might have personal factors that I cannot account for (like lack of funding, two body problem, etc), but in general, I would advise any student of mine to take all the possible/necessary time, assuming that they want to get a TT position somewhere good. If you want to teach in the back of beyond, or go to the industry (in a non-research position) then ok, rushing can work.

TLDR; If you think you are good enough now, you could be even better in a few years, which would greatly increase your chances for a good TT position.

For advisors, there may be a big change in how grad students and postdocs are paid, and what charges are being assessed to their grants. These may be helpful or hurtful to your chances of getting a postdoc position.

For instance, your advisor may be currently paying tuition charges that he would not have to pay if you were to become a postdoc. On the other hand, he may have to pay you a substantially higher salary as well as overhead to support you as a postdoc. So it's really a factor of what the rules are at your school.