This is one of the classic scenarios for which the conjugal sponsorship process was created, basically.
As I noted, my usual 'suggestion' is - where possible - to get married, because it responds to at least the legal question. Which can be particularly important for some cases, like where the 'barriers' question is not so clearcut.
BUT: 'possible' I mean here in the sense of practical or feasible, and I don't have an answer for that, and I don't intend for the 'usual' suggestion to be some universal rule (particularly for those who simply cannot do it). You have to decide (with your partner). To me, the fact (I assume) that neither of you currently have visas (or similar) to the jurisdictions where you could get married is strong demonstration that it's not possible (at least at present).
So I don't see any strong reason not to apply for the conjugal process now. If you both had UK/EU visas and could travel there next month, I'd probably lean towards 'get married and then apply.' If your partner doesn't have any such visa and no clear prospect of getting one, the conclusion seems clear. Note, conjugal sponsorship does seem to take somewhat longer, if only because there is [arguably] more potential for fraud - but nonetheless some clear cases get handled reasonably quickly.
[Worth noting that if it should come to pass that you can get married somewhere, i.e. all of a sudden you both had visas, you could do that later. I'm not aware of any cases where a conjugal couple got married and then were refused for the conjugal sponsorship - after sharing that info with IRCC - even though they would technically no longer meet the requirements of the conjugal sponsorship. Anyway, cross that bridge when it comes to it.]
I highly appreciate your insights.
Your assumption is correct. None of us currently have visas to the UK/EU or any other country that would allow same sex marriage.
Also to provide some more context to my partner, he is unemployed in the home country. He is a full time undergraduate student going to university, and he is only planning to commence his internship next year. He is unable to commence employment at the moment, as he has full time lectures. This was one of the strong reasons for a Canada visitor visa refusal. A refusal reason states "your current employment situation does not show that you are financially established in your country of residence".
So even if we apply for a UK/EU visa I am very much certain this same reason would make them refuse as well. Further, since he already has 2 visitor visa rejections from Canada, that would be factored in as well in their decision.
Although I am financially established in Canada, it is not good enough for Canada to grant him a visitor visa. They need to see him to be established in the home country, to make sure he will go back at the end of the visit.
So as you rightfully said, there is no clear prospect of getting a UK/EU visa at the moment.
When we applied for a visitor visa to Canada, we mentioned the purpose of visit as tourism. We didn't mention we want to be married when he is here. This is because a lawyer advised if we mention the purpose of visit as marriage, they might refuse the visitor visa thinking he would not go back. So we prepared an itinerary for his visit, and I mentioned on my letter of invitation that I will ensure he doesn't overstay his visa. But this was still not good enough, and he was refused.
I have a question at this point, because you have been stressing so much on marriage. Would you now advise me to go ahead and apply a visitor visa for a 3rd time explicitly stating we want to be married, and file an inland spousal sponsorship? If we honestly say that is our plan, do you think IRCC would approve a visitor visa? As per the lawyer we spoke with that would still not be good enough, as my partner needs to be established in his home country, and show he will return at the end of the visit. If we state that he has plans to immigrate, his visitor visa would be outright rejected from what I heard. I like to hear your opinion on this.
Also, in our conjugal sponsorship application do you think we should talk about how Canada refused him entry on a visitor visa, which made it impossible to carry out our unmentioned plan of marriage, and file an inland sponsorship? I feel like there is a conflict here because the original purpose of the visitor visa was tourism.
Do you think we should also talk about how none of the UK/EU countries also would not let him in due to his employment situation?