These agricultural products paid to the emperor were used to finance public projects.
1.首先确定“别人的讲解是：这个句子很完整,paid to the emperor 是修饰前面的 these agricultural products” 是完全正确的.这是因为 paid to the emperor 属于过去分词短语作后置定语,这是分词短语的基本语法功能之一,相当于定语从句 that were paid to the emperor,即：
These agricultural products (that were) paid to the emperor were used to finance public projects
因此仅仅认为省略了关系词 that 是不对的.如果非要说省略,只能说这里不仅省略了关系代词 that,同时也省略了构成被动语态的助动词 were.
2.你把句子的意群划分为：these agricultural products （paid to the emperor） were used to finance public projects,是对的.如果“省略修饰部分” paid to the emperor,当然可以“暂时把它看成”：
These agricultural products were used to finance public projects.
那么,这就更加证明原句是个完整的句子,并不存在省略 that 的说法.