Multiple Canadian passport? Is it possible?
 

Canada does issue a second passport, the company I work for lobbied for months and the govt finally relented.
When attempting to obtain long term work visas in Angola, workers were having the same problem you were facing. So several people I know have a second passport valid for either 2 or 3 years. These passports were issued in Halifax and St. John's.

Getting a second passport is no problem at all. Just follow EXACTLY this procedure,

Get a letter from your company, complete with clear, easy to use contact details for the company and a name for the contact, describing in detail the reason you will need a second passport and be sure they include the dates you plan to travel, and the countries you will need to visit, the country that will be holding your passport and why, and any other information you think Passport Canada might need. When you have that letter download a passport application, not a renewal application, you need to follow the steps required for a first passport, get your passport photos and go visit the passport office. If all is in order you should get your second passport in less than a week. This is what I did to get a second passport.

Worse than getting a Russian visa?

Many years ago, China would not allow entry with a Taiwanese visa in your passport,
and Taiwan would not allow entry with a Chinese visa ( This was before China "opened up", and there were very few of us travelling there). What we did was arrange for a
second Canadian passport that was held at the Canadian High Commission in HKG - at
that time all flights in and out of China were via HKG - go to HKG for 2-3 days, exchange passports in HKG, apply for rush Chinese visa, go to China, return to HKG,
and then exchange the passports again for the originals.

Those were the good old days. I also recall having to get my Taiwanese visas in Seoul,
when what is now the Chinese embassy near the Chosun Hotel was then the Taiwanese
embassy. It was a great step forward when we were finally able to send our passports
to USA to get Taiwanese visas in advance, then finally there was a Taiwanese trade
office in Canada that was authorised to issue visas.

Multiple Canadian passport? Is it possible?
 

A number of islamic countries still do not allow you to enter when your passport has an Israeli stamp in it, a second passport is the easiest way to avoid this issue.

Wouldn't it be easier if they attached a visa that could be removed upon departure, similar to the Cuban tourist cards?

Often, when arriving at Ben Gurion, you will be asked before your passport is stamped if you want it stamped (or have the stamp put on a slip of paper [along with visa type, expiry and a copy of your photo]).

Israel now just puts a slip of paper with entry details in your passport rather than
stamping it.

Of course that does nothing if you are crossing by land, as the exit/entrance stamp from the country bordering Israel gives away the fact that you were there.

You are allowed to have two passports. My passport was stamped at the Arava Border Entry point in Israel, which would cause problems with entry to other Islamic countries.

When I went to the Toronto Passport Office to get a new Passport I was offered the option of getting a 2nd one with the same validity and no additional fee on top of the normal processing fee.

The condition is you can only have one in your hands at any given time. You have to go in person to the Office and swap them when you need the other. I didn't ask if it could be possible to have both out if one is tied up getting visas.

Oh, not limited those. A few years ago, at a land border Canada into USA, travelling with an Israeli in the car, the agent leafed through my Canadian passport, saw several pages with Arabic stamps and pointing at one “Is that Iraq?” (Pronounced Eye-Rack). “No, sir, that is Lebanon.” Eventually, “I advise you to withdraw your application to enter my country or I will process it as denied entry”.

Russian visa is a breeze compared to Saudi or India. the documentation req'ment for Saudi is ridiculous and changes by the hour. I got my russian visa in 3 days .. I sent the exact visa app to the saudi consulate as i have done in the past .. after 2 weeks it came back asking for more stuff.

OP : if you can get a second passport from a different country (e.g. if your parentage makes you eligible for an EU one), you can use that to get into the US while the Canada one is tied up.

I dropped my passport at the Brampton BLS office for my Indian visa last Monday, and it was ready by Friday. It shouldn't take 2 to 3 weeks.

Ah! I got similar experience...I filled the request, got the invitation letter, got my company letter submit then needed the company bank account stamped by board of commerce, then need something else...anyway...19 business days it took...and half of those required papers weren't mentioned on their site...