Maybe someone else who knows more about montauk daisies will chime in, but in the meanwhile, I'll give my 2 cents.

Bugs are a normal part of gardens. If they're not doing any harm, and/or if you haven't identified them and determined that they are Evil, just ignore them.

It's perfectly normal for the old stems on MD to be brown, so that's not really an indication that anything is wrong.

If you still have the plant out of the soil, post a photo somewhere and maybe we can ID the bugs. There's a chance that they're completely benign and that your MDs are too.

Best way to propagate MD is by softwood cuttings. They root very easily. I used to have a one of those but never had bugs. I say burned the roots and take some cuttings if the top part of the plant doesn't has bugs. Hope this helps.