Corbyn had popular support but, like Miliband, relied relatively heavily on unions...and one union at that: Unite (and btw, his popular support was always overstated hugely...in late 2019, you had a sizeable minority who thought he would walk it, he was massively popular in the Westminster bubble...this is despite him being regarded as a totally odious figure under Blair when he was a backbencher).
The Tories have never had this issue because their electorate is relatively diffuse, and MPs have been quite willing to stab their leader in the back at the first sign of trouble.
Blair (like Thatcher) was an accident. I agree with your point but the Tories have been generally able to produce more effective leaders with their constitution.