Site.com will be completely separate from the community builder going forward infact from Spring 16 release .Community builder is the future
The community builder now has more flexibility to build pages .Lets take a look at possible options now using Napili to sort of completely replace UI
1)You can create your own lightning template page .It can be either from the object or you can choose from assigned Templates

2)There are options to override the Navigation and header if you want to build your own

3)Since the standard template is all lightning based you can use community builder and replace any standard components with your own

4)There is an option to sparingly use your own CSS overrides for existing standard components

5)Apart from the layout options provided I have mentioned in step 1 ,if you still want to customize further ,you can build your own layouts.You will need to implement an interface known as forceCommunity:layout and add your styles .
Here is a quick sample code to help you with
<aura:component implements="forceCommunity:layout" access="global">
<aura:attribute name="column1" type="Aura.Component[]" required="false"> </aura:attribute>
<aura:attribute name="column2" type="Aura.Component[]" required="false"> </aura:attribute>
<div class="container">
<div class='contentPanel'>
<div class='left'>
{!v.column1}
</div>
<div class='right'>
{!v.column2}
</div>
</div>
</div>
</aura:component>
With all those toolset provide you should be able to completely customize to what you need and sky is limit with community templates going forward .All you need is skills to build lightning components