Probably not actually, the only reason Rome survived the fall was because of the Church drew people there. Its population dwindled almost 90% and was sacked multiple times in-between the fall of the Western Empire and the Medieval period.

And even then the power of the city waxed and waned depending on who the Holy Roman Emperor was. The Holy See was moved for 70 years to France in the 14th Century

They always had other stuff to spend it on, like Armies, gifts and tributes to avoid attack, defenses, feeding the city, building new buildings with materials taken from those ruins. Keep a pagan building up to its prechristian status was not something anyone would want to spend money on.