I doubt the wrong intake gasket would cause the heater to quit working. If that were true, you would have other, more obvious problems than the heater. Like water in the oil or a vacuum leak.
5 minutes to get the engine to temp sounds about right.
Getting the heater to work is really pretty basic and has few moving parts.Let's go back to the beginning and see if we can't do a diagnostic and pinpoint your heater problem. Please don't be offended but this is going to get really basic but don't skip a step. I know I sometimes have to backup and start at step one.

The engine cooling system seems to be working correctly. Warming up to temperature and not overheating.
We still haven't confirmed if there is actually coolant circulating through the heater core while the engine is running. (warm, not hot hoses) Or where your hoses are connected. Until we actually know this, everything is just a guess on our part. You have to help us here by being very specific with your answers.
We know the heater core is new and clear.
Did you replace the heater hoses? Are they running clear? No kinks or crud plugs.
Is the hose fitting into the intake clear of blockage/restrictions?
Here's one way to check coolant flow through the entire heater system without disconnecting anything.
If you have the hose connected to the radiator tank it is a simple check. It can even be done while the engine is cold.
Drain enough coolant out to get the level below the heater hose connection on the radiator.
Leaving the cap off, start the engine.
Look into the radiator filler neck and see if you are getting any coolant coming from the heater hose outlet.
There should be a strong stream coming from the heater into the radiator at all times. Even when cold and the heater turned off.
A dribble means a partial blockage and no flow means a complete blockage or the hoses are hooked up incorrectly.
If your return is going to the water pump, it's a little more involved but not as much work as you have already done.. You will need a buddy to help.
Do this while the engine is cold to prevent burns.
Disconnect and plug the w/p heater hose port.
Run the hose from the heater core to the radiator filler neck or a bucket.
Refill the system but leave some room in the radiator for coolant from the heater.
Have your buddy start the engine while you hold the heater hose into the radiator fill or the bucket and watch for coolant flow. (Keep the end of the hose higher than the top of the engine. It should only take a few seconds to see if there is any circulation.
If little or no flow, check each individual item starting at the heater hose connection to the engine. Fitting, hose, core, hose. Eliminate each piece as a problem along the system. If you have a good flow we can move on to other alternatives.
Keep us posted on your progress.