There's actually a fascinating shaila of why we don't bring a korban pesach bizman hateh. By korban pesach we pasken that hakrava is not meakev and you don't need a mizbeach. All it need is shchita b'eazara. Since Tuma mes is hutra bitzibur, and one can easily remedy the other tumos, we technically can and should be bringing a korban pesach bizman hazeh.
Now while there might be external reasons for us not to go on har habayis even though there is no issur medina, and I personally strongly dislike the fact that people go up bizman hazeh, if we take our personal feelings out for a minute and look at the facts, in Jewish history we never see that they stopped being makriv korbanos for fear of incitement. It's truly an interesting shaila for those who are willing to look into all the halachos involved