Honestly, if I had $2k budgeted for single-shot rifles in .223 and 30-06 (fine choices for a pair of working rifles!), I'd find two used #1s in good condition. I'd put glass I liked on top (probably an old T6 Weaver on the .223, and a 4x or 6x from Weaver or Leopold on the '06), fit backup iron sights to the '06, and use the left-over money for ammo. Then I'd make them mine by trying to wear them out hunting--but that's me.

For those cartridges, you might look for a couple of the recent High Walls from Browning or Winchester. There are variations and different models, meaning different internals, but they seem decent rifles. Parts availability is about like any discontinued rifle, but the maker (I think both of these were made by Miroku) made a decent base rifle. I'm not sure you could get them both for under $2k, but you might be close.