So when a run is made on a server with no ICE protecting it, the access phase immediately begins. I want to clarify, can the runner still choose not to access cards at this point and end the run?

When running an unprotected server you jump to step 4 which does allow you to jack out before accessing any of the servers cards.

However, doing so also prevents the run from being considered successful.

To clarify above. If you declare the run to be successful (IE, you don't jack-out). You MUST access the cards, you have no choice.

I had it in my head that there was no opportunity to jack out if you ran on an undefended server, but it looks like there's a razor thin window.

There is a Paid Ability window before declaration is successful, so in theory the corp could do something with an installed card that might change your mind.

Example:
Gabe is running against Jinteki who has no cards in HQ and no ice on HQ. There is a rezzed Panic Button in the root of HQ. In step 4.1, Jinteki activates Panic Button twice to draw two cards, then smiles broadly. Seeing that Jinteki has 8 credits, Gabe decides the risk of hitting a Snare! is too great, and jacks out.

Now, given that there is a Paid Ability and a Rez Cards Window between declaring a successful run and access, it's entirely possible in the above example that Jinteki might wait until Gabe has fully committed before trying to draw into a Snare!. If the Panic Button was unrezzed, they would have to wait until Gabe had declared the run successful before rezzing it and paying for cards.

In a game last night, I had an undefended remote with ChiLo City Grid, Bernice Mai, and a SanSan City Grid, and it was after he'd committed to accessing that I rezzed the two Upgrades, ran the trace, gave him two tags, and then he accessed. He trashed the SanSan, and at that point he had spent 5 credits and would need two clicks and 4 credits to clear the tags, so I considered the tempo tradeoff an acceptable outcome. I'd have preferred to have the SanSan on my turn to score an Astroscript from hand, but I try and set up situations where I can take advantage of either option the runner chooses.

*edit* yep, misplay to have 2 region in the same server. Listen to what I mean, not what I say.

In a game last night, I had an undefended remote with ChiLo City Grid, Bernice Mai, and a SanSan City Grid,

you can't have 2 region card in the same server...

Oops. Got carried away, I guess.