This might sound a bit silly, but "that's the Torah and I need to learn it" (R"A).
A woman can be Mekudeshet in three ways which Halachah-wise are all equal (neither makes a better Kiddushin).
A messenger can do the work on behalf of the sender either by money or a bill, but not ביאה (Kesubos 74a).
We also hold that there does not have to be proof of the מעשה, only ראוי למעשה, so the witnesses only see them entering and exiting (and zipping up). So theoretically we could allow the messenger to seclude with the bride on behalf of the groom without doing the מעשה.
To remind that that ביאה is for Kiddushin only and is different from the second ביאה for Nissuim.
I need a source that speaks explicitly about why not the last. Thank you.