The reason the eyes of the elder son aren't a Rinnegan ( or at least doesn't look like one ) is that he lacks the Yang part and vitality of his brother.
Same thing goes for the Younger son or Hashirama ( since they are equal imo ) , their Mokuton is not as powerful as it should be because they lack the Yin part and the powerful chakra of the Elder son.
Note : Elder son is Yin oriented ( thought he does have a Yang part but is very weak compared to his brother ).
Younger Son is Yang oriented ( thought he does have a Yin part but very weak compared to his brother )
The two sons are both weaker than their father despite having Max Yin , little Yang ( Elder ) and Max Yang , little Yin ( Younger )
Their father had a Perfect Yin/Yang ratio.
You get my point ? Maximizing Yin or Yang will never make it better than a perfect ratio.
Now we have seen Dojutsu getting more powerful by strong Yang entities ( Yang Kurama empowered Sasuke , and Kakashi )
Hashirama's cells empowers the Sharingan like Shisui's MS ( reducing the charge time from 10 years to a day ) , Obito's MS ( Kamui without blindness ) , Madara's EMS ( evolved it into Rinnegan )
And so it should work the same way for the opposite side ( Powerful Chakra of the Yin+ High Chakra reserve of the Yang part would perfect the Mokuton )
In short Madara didn't reach the pinnacle of the Elder son's power and that's why he had to use a Bijuu to force a very near stalemate with Hashirama.
Now why Madara didn't reach his full potential ?
We don't have a response from the author to that but my opinion is this :
SO6P knew that his elder son's idea of peace is wrong so he programmed a limitation to the descendants of the latter and that limitation is :
The elder son's descendants will never reach their full potential if they are full of hatred and have the same ideal as their ancestor.
And if they dropped hatred and ideals and followed the ideal of the Younger son , then they would be able to reach their full potential.
( He couldn't do the same to his elder son because he already reached his full potential and it was too late for him ).
I'll explain what i just said with Manga facts :
Madara said : " If the two opposing forces cooperates , real happiness can be obtained. "
In case of that version of Madara and Hashirama that would have never worked , because even though Madara didn't have his hatred toward Hashirama anymore , he still had the same ideals as the Elder son here ( cooperation is nothing but a silent fight ) ( while i'll walk towards my real dream ) :
And of course he still deep inside hate Tobirama for what he did.
And so that's what stopped him from reaching his full potential and cooperates for real with Hashirama.