The bar for needing an assessment of needs is in fact very low. It is as follows: whether the child has or may have a SEN. And whether they nay need Special educational provision to meet that need.
If the answer is yes to those two questions then the LA must carry out a needs assessment.
You do not need expert reports or to have spent a certain amount of money to meet this requirement.
Let's just break it down a bit more, what is a SEN? By definition it is a learning difficulty which results in significant greater difficulty in learning than the majority of others the same age, or a disability which makes it difficult for them to use the facilities normally provided for others.
In your sons case, he has a speech delay which from what you've said has hindered his ability to learn as you would expect for a child of his age.
The second part is whether that SEN may need Special educational provision. What that means is : education or training that is additional, or different from, that made generally available for children of the same age.
These are the only two hoops you need to jump through for an LA to have to complete an assessment of needs. So focus on the proof for each. In the case of the the SEN, ask the school for his attainment data, have you had any speech and language therapy done in the past, what did they say ? If he has made progress is it as a result of intervention only.
Then look at SEN provision. Have the school done anything to try and help? Intervention sessions with TAs or changing the way they approach his teaching.
SENcos often don't know the law. They know what the LAs tell them. It's not always the case of course. But it's certainly fair to say your SENCO is misinformed, so you need to read up. IPSEA is a great start.
I recommend you have a good look round the IPSEA website. There's a whole section on needs assessments. Also, download chapter 9 of the SEN code of practice. (Google it, its on a government website) as this spells out the law in the form of a slightly easier to read guideline format.