The 3Ohm impedance is definitely odd and a bit of a challenge, though for some reason I doubt they are really that hard to drive - if they were I don't think an inexpensive HTIB amp would be able to drive them. My guess is either of the receivers you mention would be able to drive them to reasonable levels.
You are correct however that most receivers can not power a passive sub - they have an RCA output intended to go to a powered sub-woofer. That's not saying you can't use passive sub(s) but it does mean you will need another amplifier for it (them). In fact lots of people on the DIY forum have passive subs powered by pro-audio amps. I have two DIY 15" subs powered by a Behringer iNuke 3000D amp.
Since you on on a budget you have 4 options with the sub:
1. Buy a quality subwoofer to use with your current and future system. Your room size and listening preferences will have a lot to do with how much "sub" you will need. To get to real quality subs that can go to 20Hz you need about $500 - My recommendation is the Monoprice Monolith 10" THX Ported Sub. Realizing that $500 may be more then you want to pay, I think the RSL Speedwoofer 10S is the best $400 sub out there and the BIC Acoustech PL200-II is the best $300 sub. Neither of those last two will get you much output at 20Hz but they are both good subs at their respective price points.
2.) Buy a plate amp and cut a hole in your existing sub to install, making it a "powered subwoofer". Parts Express has a Dayton Audio 70watt subwoofer plate amp for $50.
3.) Buy a cheap amplifier and hook it up to your passive sub. Most passive sub users use either Behringer or Crown amps with the budget leader being Behringer. The cheapest amp is about $250 though so it's still an investment. Parts Express does sell a "Pyle" brand amp for $100 and that might be a decent stop gap for a while.
4.) Buy a cheap "temporary" replacement sub. Most of the subs under $300 aren't great performers, but one could get you by until you upgrade the rest of your system. Parts Express has a Dayton Audio 10" sub for $120. I know nothing about it, and am sure it's not the worlds best sub, but as a budget stop gap, it might be worth a shot.
Yamaha receivers are generally thought of as the most reliable out there at the moment, and generally sound pretty good. I haven't had a Yamaha for a decade, but there is no particular reason that would make me not buy another one. I think that sounds like a good option.
Sony doesn't put a lot of money in their home audio group anymore, but used to have some great stuff in their ES line. I'm sure the Sony would be a fine unit too, if it has the features you want.
7.1.4 Theater Room (In Progress): JVC-RS46U, Silver Ticket AT 2.35:1 142”, Onkyo RZ830, Anthem PVA-7, Panasonic DBT-110, JBL Studio 270’s, 4 - 15" DIY Sealed Subs
3.1 Living Room: Samsung 64” F8500 Plasma, Anthem MRX 300, Dynaudio Audience 52’s, Dynaudio Audience 122C, NHT SubOne