Tomb of Horrors appears to be the odd one out in terms of published adventures, originally designed very specifically as a challenge to his own group.

Gary Gygax himself said "There were several very expert players in my campaign, and this was meant as yet another challenge to their skill—and the persistence of their theretofore-invincible characters" The cover of the original ToH also states it was designed for tournament play, which indicates a far more competitive requirement than a normal adventure module.

I'd very much suggest comparing ToH to the style of, say, Temple of Elemental Evil which has much more of a balance with narrative and roleplay aspects when you read through it.

One thing to bear in mind is that his "style" was likely that of any good DM - to constantly adapt his style to fit the situation at hand. Given players that defeated everything he placed in their way with relative ease, he ramped up the difficulty. That doesn't mean he'd necessarily throw that level of difficulty at a different group. Any good adventure should be designed with a specific aim in mind, and a good author will tailor their work to that aim, rather than their personal gaming style.

This kind of approach can be seen in passages from the AD&D 1E Dungeon Master's Guide, such as "The testing grounds for novice adventurers must be kept to a difficulty factor which encourages rather than discourages players", and "If things are too easy, then there is no challenge, and boredom sets in after one or two games" - to me, that pretty much reads as "be a tough DM when your group needs one" and it seems he felt his group needed one. On the RP side, he clearly states "Place regular people, some 'different' and unusual types, and a few non-player characters (NPCs) in the various dwellings and places of business. Note vital information particular to each" and "When they arrive, you will be ready to take on the persona of the settlement as a whole, as well as that of each individual therein" so I'm pretty certain he didn't feel RP was something to be ignored.

Any quotes from players in his games should be taken with a pinch of salt, as they're undoubtedly going to apply to how he played with that particular group.

tl;dr: His style appears to have been to present the game that would best entertain and engage his players, rather than to play the game "his way".

I couldn't find a source that traced Robilar's progress room by room through the Tomb of Horrors, but according to the Dragonsfoot forum thread:

Robilar losing all of his orc flunkies at the onset, in the initial passage. When he found the tomb of the demi-lich Robilar scooped all the magical treasures he could into his bag of holding and ran off leaving the demi-lich hanging. [sic]

In other words, it seems Robilar took with him orc minions to activate the tomb's multitudinous traps, and he was careful to leave undisturbed anything suspicious, such as a creepy bejeweled skull.

Robilar's own account

On his site "Lake Geneva Original RPG Campaign" author and game designer Rob Kuntz, Robilar's player, was asked by user Thorsten S. about Robilar's adventures in the tomb. Kuntz was kind enough to reply in the site's comments, saying first that

I [Robilar] was careful and proceeded slowly; when I came upon the demi-lich I merely hefted the nearby treasure and ran out of the room, which seemed to perplex Gary. I'm not so sure of what more to tell as it happened very long ago, so the main points must suffice.…

Then, in another comment, Kuntz offers a few more details, saying

Gary called me and said he had a new level to playtest. We used to playtest these between us upon one of us finishing a level. Both of us always suspected the worst, as we had always thrown big curves at each other and our players over a year's worth of solid day-to-day playtests and play.

Five of the five orcs died at the entrance, one whom Robilar dispatched because it refused my order to advance into the entry corridor, the other four from the pits within said corridor.

I don't recall using this stratagem before; it's a left-over tactic from wargaming days.

The rest of it is vague. This was an initial playtest, so Gary may have changed parts for the final published form….

Kuntz concludes in another comment:

I do recall the face [that] I probed with a pole, the latter which came up an inch or two short afterwards. That and the pits… let me know that this was a trapped based adventure. Thus I slowed all processes of exploration accordingly—i.e., I was doubly suspicious of everything.

…As anyone should be when exploring the Tomb of Horrors!

Special thanks to Thorsten S. for his assistance in generating this answer.