Recently naturalized American visiting the UK, previously denied visa. Apply for entry clearance?

You are a recently naturalised American (congratulations) and your immigration history includes a refusal from when you were (presumably) a visa-national. You did not include a scan of your refusal notice and it's not relevant at this point.

Your description of the refusal tells us that they got you on 'funds parking', this is a phrase coined exclusively here at Travel Stack Exchange to describe a situation where an applicant's source of funds has a dubious provenance. They refuse on those grounds all the time. You think that you were not given an opportunity to explain, but you were: it's called a fresh application; if you had included documentation showing the lawful provenance of your money, you would have had great chances for success.

You want to enter the UK through the juxtaposed controls in France. That's fine. There's nothing preventing it. You can also take the Eurostar if you want, or you can fly from France, or whatever. These strategies are fine, but they do not improve over simply flying from the US. If one inward route were easier than the others the Chief Inspector would spot it and it would be tightened up. If anything transiting the juxtaposed controls is more rigorous, we have two cases in our archives here where Americans failed that route.

Whichever way you pick, it's about the same, all in all.

Is this better than applying for entry clearance ahead of time, bearing in mind obtaining entry clearance ahead of time does not mean I will automatically be allowed entry into the U.K.

Entry clearances get revoked if the person used fraud or deception or if there's an Interpol alert for the person. They are never revoked on a whim or in a way that second guesses the judgement of a British Consulate General abroad. It simply does not happen. Instead, they are the most sure-fire, stress-free way of entering the UK; there are millions of entry clearance end-users every year.

User Michael Hampton (to whom thanks and a tip of the hat) has given you the recipe for never having to explain your refusal going forward...

It will be much easier and less stressful if you apply for the entry clearance beforehand, this time including all the appropriate documentation. Having it approved will also effectively deal with the previous refusal and virtually eliminate any chance of problems at the border.

Good advice. And the same cannot be said for having obtained leave-to-enter at a border crossing.

Notes:

I like the comment from "MadHatter" below...

"...but because if it all goes pear-shaped he'll still be in France and able to enjoy a holiday, rather than being detained at the airport and thence flown back to the US..."

If the possibility of a removal seems nascent, and if arriving from America, a solicitor's advice would likely be to carry confirmed onward travel arrangements to France (or anywhere else outside the Common Travel Area) and if it all goes pear-shaped to seek admission in another category, viz. a temporary admission or perhaps more appropriately, one of the transit categories. The trick is to have your stuff ready and know what to apply for if leave-to-enter fails. Presence of mind can be very helpful in these situations.

First of all:

I assume as an American I am allowed to travel visa free to the UK, despite my previous visa refusal

This is correct.

I am married, have a well paying job as an engineer here in the USA, have traveled several times to the UK and never stayed more than ten days [...] So basically I want to travel with all my financial documents plus copy of previous refusal as if I were applying for a visa, and present my case if questioned at entry.

The way I see it, you should be fine, though possibly with some delays.

The GOV.UK website does say that you should bring the same documents to the border that you would send to the visa application centre. Although this is an exaggeration in most cases, you will want to take their word for it.

That said, if you collect all available proof of your ties to the US in an A4 soft binder (not just financial ties, but also employment contract, marriage certificate and anything else) and bring them to the border to show if requested (don't present the pile of documents upfront), you should be let in.

That said, an entry clearance is and always will be the safer option, because, as others have pointed out, not only is it a pre-approval for entry (unlike US visas, which only let you apply for admission), it will also nullify the circumstances that led to your previous visa refusal.

So, in summary: yes, you should be fine to travel without a visa if bringing all above mentioned documentation, but to play it really safe, you may want to get a visa.