It's not JUST average current Nigel, you pulse too much current through a P/N junction the junction will vaporize outright, the heat can't dissipate fast enough, even if the average is within it's tolerances. But I didn't know remotes used 1 amp pulses. I've always been afraid to do anything over 200ma's for any LED, but I believe ya =) If the transistors can handle it it's the easiest sollution.
Latching shift registers is the other, but that requires more IC's. They're pretty simple Dan, you shift bits into the registers one at a time via a clock line, and then when you trigger a certain I/O line those bits are all pushed out onto the IC So you can update a theoretically infinite number of I/O lines by chaining them together, it's limited only by the speed you can clock the serial data in. I dug around on Digikey and found this one.
https://www.electro-tech-online.com/custompdfs/2009/03/397-1.pdf
But that's just the first one I found.
For 70 elements (You're doing 10 7 segments right?) you'd need 9 of them.
Which would put you out all of 5 bucks total.