How to temporarily cover a hole in the roof

I agree with Ed Beal. But yes, there are 2 ways to do this & they're equal in strength, for what's planning to be done. However, get a Permit from the Building Dept., so they can approve or correct the design, fasteners & supporting post(s) dimensions & footings. Plus, they can record that the patio was converted to a porch.

1st way is to do ledgers on both fasciae into structural members. Typically you'd pre-build the new structure so the short rafters are end screwed or nailed through the blue-line's ledger & the long lower beam & top ledger are through the side ledger (rafter). Also, use joist hanger type connectors to ensure nothing can pull away from each other.

2nd way is really the proper way & is likely what the one contractor was talking about. You'd remove both fasciae & screw or nail directly into the structure. For the slope side this is much less important. But, the blue-line top would have the new rafters cantilevered or sistered onto the home's existing rafters. This is much more secure & even lets you re-use both fasciae.

judging by the shape and cut of the wood, the method of assembly and the size of the members, i would assume you have a tile roof? either way, it looks like it was built for a very heavy roof load.

the rot you are seeing is just part of the problem, it most likely extends far into the timber and may even be present in the adjacent rafters.

bracing the beam (thats the transverse thing under the rafter) may not help much as it looks rotten as well.

the best thing to do is put in a post. this is a vertical column from ground to the bottom of the rafter. put down a wide block of wood or a concrete slab first to spread the load over the paving stones below. use a metal bracket at both ends and concrete anchors, lag bolts or nails to secure the post to the base and to the rafter. if you use a good solid post and are careful in your work, it should last until you can replace the roof.