Hm. I'm not buying Herron's argument that what was good for Josephus and the Mishnah regarding the use of the present tense is
not good for 1 Clement, at least regarding "One simply cannot say that of 1 Clement: we do
not know that 1 Clement knows the Temple is in ruins!"
Alright, we don't "know" that 1 Clement is aware that "the Temple is in ruins," but for the various reasons I've given I
suspect that it was written c. 95 CE (or at least after Paul and before Hegesippus), so I at least feel comfortable with the
idea that 1 Clement knows that "the Temple is in ruins" like Josephus and the Mishnah.
And in the big picture it looks like I'm with Lightfoot on this (to judge from Herron's takedown, but I want to see what more I can find about him on this), but I'm not in a position to say (based on the grammar) if I can agree with Herron that Lightfoot's argument is weak.
On the question of whether there were sacrifices on the Temple Mount after 70 CE, you make a good point that 1 Clement does say the altar was in front of the sanctuary and I need to give that more thought. But perhaps it was there in a "crumbled state" as per Clark (which Herron doesn't buy). And Herron makes a good point that the skeletons that were found by the Temple Mount would have rendered the area unclean and would have compelled Jews to bury them and that they didn't indicates that they thus didn't offer sacrifices.
But if the bodies were in caves and a drain under a house, would post-70 CE Jews have been aware that they were there (particularly if everything was in a "crumbled state")? And I don't know if it is the same thing Herron (via Kenyon) is referring to (it seems more recent), but there appears to be some question about the dating and identity of a mass grave of skeletons found near the Temple Mount, e,g,:
Senior archaeologists approached by Israel HaYom with photographs of the skeletons said that they were not enough to determine the history of the cave. Samples need to be taken from the site and dated, they said, before deciding that the mass grave holds Jews and not Muslims.
http://www.israelnationalnews.com/News/News.aspx/158417
Maybe the same applies to other skeletons. But I need to look into this issue more.
And where have I read that the Temple Mount is sanctified for all time, Temple of no Temple? Perhaps that could be a factor in the question of whether sacrifices were made there post-70 CE.