So why didn't Shepherd.

I say it's probably because they wanted to give the player the option of shooting Shiala in the head.  If the main plot became contingent on Shiala melding with the Asari councillor, that option could not exist since it would defeat the plot... so, instead they made it such that, if spared, Shiala simply makes a choice that involves staying on Feros rather than joining Shepard and returning to the Council.  Sure, it would have been better written if they didn't make the ME1 plot rely so heavily on the Council not believing Shepard... making the Council out as ignorant jerks just for the H of it.

Well if it could prove something.

Actually, Tevos is very supportive of your endeavor, though subtly so. She can't be outright combative of the other two who clearly don't believe in Shepard.

Oops... Ctrl-C didn't take.  That link was for another post.  I've corrected it now.  Sorry.

I imagine there's a strict convention of "mind melding isn't considered credible or admissible testimony under any circumstances, ever" that the Citadel must have inherited from Asari culture.

It's not only supposed to be far too intimate for any professional setting, the visions we see through it in the series are also very abstract and emotionally charged, and not always even based on objective reality. Any kind of barrister or judge who accepted such a testimony would immediately be open to accusations of having been emotionally compromised or having based their verdicts on unreliable data, neither of which anybody would be able to either verify or rule out.

Even worse, for the Asari Councilor to personally accept such a connection with one of Saren's accomplices could totally undermine her trustworthiness and cause a rift between the Council members. Who's to say she wasn't in on the conspiracy from the beginning, and is using this yucky Asari practice to manufacture evidence for whatever reason?

Shepard accepting it as a way to follow up on a lead is one thing, but throwing it around as a basis for policy decisions in the most powerful senate in the galaxy...

1) Uhm, no I'm not, I'm assuming that the Council members need to be able to explain their actions to each other, and to their voters if a conflict between them requiring intervention emerges. If two out of a triumvirate can't trust that the third isn't compromised by terrorists then that's not really a sustainable basis for a political body, is it?

2) Of course she did, she was presented with an audio recording proving their link to murderous robots terrorizing the galaxy. Trying to advocate for Saren and Benezia in the face of that would definitely have been political suicide, and it wouldn't have protected either.

Look, I'm not saying that she was in league with Saren or even that her accepting a mind meld would definitely have led to trouble, just that keeping mind melding far away from government decisions of any level would definitely be a cautious and appropriate general policy, and that accepting it in this instance at the very least might be construed as reckless and improper, and could potentially lead to a lot of suspicion and discomfort between the councilors. Which for any politician would mean a hard "no".

I certainly wouldn't have done it in her shoes. Much safer to just have Shepard run the extra lap around the galaxy looking for more substantial proof.

1) Sure she does. She sends a capable agent and their crew of skilled individuals and trusts them to gather information without her having to personally stick her neck out for it, which is the best you can hope for from any politician. Whether or not the issue is immediately public is irrelevant, she's still accountable for her actions and knows that she can be replaced, and doesn't want to be.

2) No, you posited that her condemning Saren and Benezia is proof enough that she's against them that the other Councillors couldn't suspect her of being compromised for accepting the mind-meld. What I'm saying is that her refusing to condemn those two in the face of hard evidence would be sufficiently moronic, suspicious and pointless even if she wanted to that no, it doesn't.

3) So? In the second game she believes the threat - the Geth in her mind - to have been contained, having no substantial reason to think otherwise. She wasn't there to hear Sovereign speak on its own accord and claim dominion over the galaxy.

To be honest I don't find Shepard's visions very compelling evidence either. They're just a series of vague and subtly gruesome images and noises. They could mean almost anything, or nothing at all. Just someone's terrible nightmare programmed into the Beacon for whatever reason. Shepard's out-of-nowhere conviction that they're proof positive of an ancient machine extermination of all life returned to haunt everyone rings really hollow in the early game, before it's confirmed by independent sources.

I don't see that delivering the vision to the Asari Councilar would particularly help anything even if she was liable to accept it. 

She's just doing her job, which is to keep Citadel space running, not solving mysteries. She and the other Councilors send Shepard to continue rooting out potential Geth problems, and there's nothing to be found. Then Shepard's ship is destroyed by a freighter that the crew testifies wasn't Geth, and could easily have been pirates or one of Shepard's enemies taking them out for personal reasons.

Then two years go by with nary a peep from the Geth and no other sign that there's anything to be worried about on that count. Humans have muscled their way into the Council and things are pretty hectic for a while, and that takes up attention. Each separate military has no doubt been warned to keep an eye out for Geth, but there's nothing. And the Council can't act on nothing.

There's no doubt that it makes them seem frustratingly stupid and uncaring sometimes, but democratic government bodies can't let their policy be erratically dictated by the vague outdated suspicions of their spies. By definition they need a lot more to take official action, even though doing so out of hand might seem like the right thing to do to you and me.

I actually thought that was a brilliant move they made in ME2. They presented the Council with a threat they officially had no way of dealing with - small colonies disappearing beyond their jurisdiction. So Shepard has to deal with organizations with nowhere near the ethical standards of a democratic government to get it fixed, and that inevitably means supporting unsavory people, and they have to decide how okay they are with that as a matter of principle.

The Alliance is sniffing at the problem too, but they aren't putting anywhere near the effort into it that Cerberus is yet. And because Cerberus is a terrorist organization, they can't even coordinate that effort with Shepard while s/he's associated with them. It's a lovely combination of practical policy conflicts and actually subtle moral boundaries that Shepard has to navigate without necessarily being prepared for it, which is very realistic when you're trying to get something done outside official channels.