I think if you've already got a degree in a field that's already a foot in the door to the kind of job you want, then going to another (expensive) degree program in a foreign country is an awful steep price to pay for networking opportunities. If you're serious about working in the US or other places, you might want to look into immegration and what's required, and your markettable skills might help you do that (e.g. working on an H1b in the US), even if you can't jump straight into the exact job you'd want, typically, there are far more openings for H1b workers in "regular" programming jobs than there are in game programming jobs. Even large game studios are small by comparison to large employers, and smaller companies usually don't need to cast an international net, nor can they usually afford to navigate the process of bringing people on a worker Visa into the country.

which should be able to get you a job in Canada

If your heart is set on the US though then you will need to look outside of games as most games companies have no problem finding staff locally and have no need to sponsor a Visa.

I'm afraid the situation in Canada isn't much different from the US. I have been through the process of obtaining a visa in Canada, and the only way to really pull it off as of recently is to specialize in a field which has a lack of local talent and to have a company that's willing to fight really hard to keep you in the country. The Canadian government will force the company in question to post a job opening for your position for a certain amount of time in an attempt to find local talent. Only if nobody can be found or if the company can make a very good case for why an applicant was not a good fit will they consider taking you through the visa process.

At that point there's a bunch of fees which the company has to be willing to pay combined with a mountain of paperwork.

For most engineering roles there's an abundance of people to fill up any open positions, so getting hired as a foreigner is not a trivial task.

Is this just for working in Canada or for permanent emigration. As software engineers and software designers are both on the express entry lists that will allow you into Canada without having a job to go to provided that you wish to emigrate permanently.

This is a different topic than permanent residency, you will still need a Labor Market Impact Assessment (LMIA) to be able to get a hold of a work permit. You can go through the express entry system without an LMIA to apply for residency, but you still need to comply with one of the immigration streams like CEC or FSW (OP might be eligible for FSW). However, since express entry uses a point based system and since an LMIA will give you a very high amount of points you need to be very highly qualified to get through express entry without one. There are some exceptions like IEC which can grant you a temporary permit (1 to 2 years), but I don't believe OP is eligible for that.

EDIT:

Just to give you an idea on the points system for express entry, without an LMIA you can earn a maximum of 600 points. Getting a score like this basically means you have a PhD, are in your mid twenties and you have years of work experience, including a bunch of other things. If you have an LMIA you immediately get 600 points added to your total score, meaning you can get up to 1200 points total. It doesn't happen too often that an express entry draw gets made which includes people with a score below 600 points. The lowest I've seen it go is very high 400s, which is already difficult to obtain.

EDIT 2: Added quote for clarity

Labor Market Impact Assessment (LMIA) ... CEC or FSW ... IEC which can grant you a temporary permit (1 to 2 years)... 600 points. ... 1200 points

Labor Market Impact Assessment (LMIA) ... CEC or FSW ... IEC which can grant you a temporary permit (1 to 2 years)... 600 points. ... 1200 points

To clarify: you're talking about Canada?

Yes, this is for Canada. My apologies if it wasn't clear.