There are some terminals to be hacked along the way, starting (I believe) in Udina's office. They all have leads about other things going on in the galaxy, some of which may be related to Saren's activities. Do you assume they aren't worth your time?
Is taking down a crime syndicate (Helena Blake's quest) not worth a Spectre's time?
You're also making some assumptions about the quality of the leads you were given.
The problem is that because of the urgency they attach to finding Saren and beating him to the conduit, there isn't any reason to go after these "may or may not be related to Saren" leads when you've got 3 "almost certainly related to Saren" leads in the main questline.
It's like trying to find a needle in a haystack, except you run off to search other haystacks in hopes that there might be a needle in them.
Taking down a crime syndicate is worth a Spectre's time, just not when that Spectre is on another time sensitive mission to save the entire galaxy.
The games present quite a few things that some of us find we need to try to ignore or headcanon around, and/or write off as bad writing. The "Race against time" could be one of them.
And speaking of headcanon, the game gives you some other tools / toys to play around with. That first Asari Matriarch writing you came across must have said something - and the game does not specify what that was. That invites you to headcanon them to say whatever you'd like. Perhaps they were numbered journal entries from a former indoctrinated agent of Sovereign (like Benezia) - IIRC, Vigil does mention that Saren / Benezia probably weren't the first. You could headcanon that they were written during the last cycle, when protheans were on Thessia, perhaps telling the asari about the harvest underway. Granted, that specific information requires metagame knowledge, but in this case it is the player's metagame knowledge rather than Shepard's. You can also be non-specific about exactly what they say, and instead choose to have them say something that Shepard would feel makes them worth looking for more.
The main thing for me is that I have to ignore the "race against time" thing because with that, there really isn't a whole lot of reason as to why Shep shouldn't just tunnel vision the main questline.
Sure having the writings be about the Reapers would be valuable information, but not if you fail your mission because you spend half the time looking around the galaxy trying to find them all. It's not like they're going to disappear if you go to find them after dealing with Saren.
Do you assume that being a Spectre means do only what the Council tells you to do, strictly following their leads? IIRC, Spectres are tasked with maintaining peace / order throughout the galaxy, and many of them act independently toward that end.
Do you suppose the Council assigned Tela Vasir to put an end to Liara's hunt for the Shadow Broker? Or is that something she took upon herself?
Shepard was given a fair bit of agency to blow them off during those comm chats. It didn't seem to me that they expected Shepard's time to be 100% devoted to chasing Saren. For all we know, they may have had a dozen other Spectres on the case. We do find out later on that they had an STG team on Virmire. Admirals Hackett and Kahoku also took some liberties in assigning other tasks to Shepard.
But at least you have the agency to make those decisions. Every time you choose to do a thing, you're prioritizing that over every other thing you could do at that time.
In order to rush off and follow only the leads you were given by the Council, your character has to: 1) Believe that Saren really does represent a clear and present danger, despite the flimsy evidence against him, 2) Believe that the leads you've been given truly are the most efficient way to track him down, 3) Believe that none of the other side leads are worthwhile, 4) Believe that none of the other problems in the galaxy are worth your time, 5) Believe that Shepard is the only agent following any of these leads or assigned to track down Saren, 5) Buy into the "race against time" urgency while looking past everything else the game offers. That's quite a lot to take on faith.
The main reason I say Shep is an idiot is because of the ignoring of the time sensitive main quest to go and do other things, not because of ignoring the Council.
The other half of the problem is that all of the dialogue up until this point backs up Shep believing that Saren is a bigger threat than the Council realizes and that he has to beat Saren to the conduit.
I suppose I could headcanon that Shep is lying to people just to get their way but then that doesn't really solve my problem: That I'm forced to change my character to fit the reasons as to why I'm doing what I'm doing, not make up the reason to fit my character.
Rather than leave it as ambiguous as possible to make it easier to do that, the dialogue feels like it forces you to go down one of a very few specific routes for RPing in order for it to make sense.