That is not a bad idea if all joists are good , I would rip a couple down to 3 1/2 and put solid blocking between the 2x10s under the 2x6s at 5 and 10 ft. Then you could just use singles if you want.
If the joists are good I would protect the top of them this time with the tape.Thanks for the quick reply and idea. Might do both. Make the lam joists and install the cross supports. I have a 3 level deck. Upper one is 12'x14', mid level when expanded will be 6'x14' and the lower level is sort of a 12'x12' with 45deg angled corners which take a 3'x3' bit out of each corner. Need to rebuild and re-locate the stair case as well. Looks like the almost 70 yr old guy has an all Summer project.