I also used Rainier - they saved my bacon when UPakWeShip fell through at the last minute (long story, best forgotten). I used their "slow boat" service, which was that they consolidated my crates with others and when they had enough to fill a container my stuff shipped. They used Sullivan Movers in San Diego on the pack end, and a local mover in the UK, Britannia (a bit less stellar, but ok) to deliver, unload, and haul it up the two flights of stairs. Because I was doing a consolidated shipment I got a discount over the full price. We were moving in high shipping season, and had basically 2 and a half lift vans (which makes 3, I guess). Everything, packing, paperwork, etc., cost me about $5,000, including full loss insurance. That was less than the UPakWeShip would have been anyway, for a self-pack and deliver to their warehouse move. (Not sure what "high season" is, but I think it is from late Spring through Summer. It should be cheaper to ship at other times of year.)
Like you, I brought things like a rock collection, 1/3 of my book collection (about 30 cases of books), my 500LPs, etc, that I could not part with. I phoned Rainier on a Wednesday afternoon, and the guys from Sullivan came on Saturday and loaded us up. That was on like 22nd of April. Our stuff got here within three days of the predicted date (mid-July), which worked out fine for us. They gave us the info on the boat it would be loaded on and we were able to track it from Long Beach through the Panama Canal and up to England.
That's very odd, the behavior you're talking about. I wonder if there's more than one Rainier shipping company out there? I worked with this guy - David Wiviott <davew@rainieros.com> and he was extremely helpful.