Yes.
First, consider that what we're really talking about here is a cow. Now bulls were sacrificed to cover the sins of a whole nation, so why a cow here? Possibly because a bull is naturally more aggressive, and this ritual requires a peaceful sacrifice.
Why does it have to be red? Possibly to symbolize that the ultimate sacrifice would become completely bloody rather than simply have its throat slit as is normal for the sacrifices.
Unlike most sacrifices, this one was not sacrificed in the Temple. It had to be taken outside of the city, just as Yeshua was.
The whole heifer was burned with ceder wood (used to build the temple; possibly also the type of wood used in the cross), a scarlet thread (which keeps appearing in the Messianic line, as when Rahab used it to mark her house when Jericho fell), and hyssop (used in the Passover; also used to cleanse lepers). So the sacrifice was connected to the temple, as Yeshua refered to His body as a temple (housing the Word of God), is connected to the Davidic line, and is connected to Passover, when Yeshua was sacrificed.
Both the one conducting the sacrifice and the one gathering the ashes were made unclean. The priests involved were certainly morally unclean, and Joseph of Aramathea and Nicodemus would have become ritually unclean when they gathered the body of Yeshua for honorable burial.
The ashes had to be stored in a clean place. Yeshua was set in a tomb that had never been used, and therefore was ritually clean.
The ashes are the only thing that can purify from the ritual uncleanness of contact with death. Yeshua is the only one who can purify us from the uncleanness of the sin which leads to death.
The cleansing ritual involves two immersions in water treated with the ashes, one on the third day, and one on the seventh. Yeshua rose on the third day, showing that the sacrifice had been accepted. The seventh day could be in reference to the cosmic sabbath (the seventh millennium) or else simply to the complete purification that we will enjoy in the World-to-Come.
There's probably more, but I'd have to go get my notes to check.
Note that none of the above takes away from the importance of having an actual red heifer before the temple can be rebuilt, which is itself a pre-requisite to Messiah's return. As you stated, God is also using the very specific nature of the red heifer as a fail-safe to keep the timing of the third temple on His own schedule.
Shalom
I have one request to make from you. For over 10 years I taught an adult Sunday school class, where we would spend a lot of time studying things like the prophetic meaning of Jewish Holy Days, Islam and Anti Christ... What I'm trying to say is that we didn't "go by the Sunday school book" but tried to have some serious studies. Last January my family moved 200 miles away and I have been kind of keeping contact with my students by posting Bible studies on my Facebook page. Do I have your permission to use the information you just gave me, and anything else I can learn from your website, for my on-line Sunday school class? TIA