First point: as a buyer, do not assume that anything an estate agent tells you is true. If he tells you the sky is blue, go and check outside for yourself. He is not working for you / paid by you, he is working for and paid by the seller of the property, so he will say whatever it takes to hook you in, and as long as he is not materially misrepresenting facts about the property (eg telling you it's got 4 bedrooms when it actually has 3) that is perfectly normal. This is not to disparage estate agents specifically - you should apply similar scepticism to anything you are told by any professional who is being paid by someone else to interact with you.
Second point: yes, some people do sell their property "off-market" but in my understanding typically this is extremely high end property and genuinely done privately: I tell the estate agent I am looking for a house like ABCD, you tell the estate agent you have a house to sell like ABCE, the estate agent tells me about your house to see if I would want it. If you look on the estate agent's website you would not see the property listed, if it is truly "off-market" (or it might be listed but with only the barest detail, "Price on application" etc.) Sometimes in this scenario you might not even have been interested in selling your house, but the estate agent knew that your house would fit my criteria and so contacted you to see if you would be interested in selling. It's basically a way for people to maintain a bit more privacy when they are selling their house.
Looking at the Martyn Gerrard website they have a number of properties listed as "sold off-market" and I think by this all they mean is that they haven't put it on Rightmove etc. Not what most people would consider "off-market", perhaps? Maybe the term is changing to mean something a bit different. Anyway, I don't think you should let the fact that something is "off-market" put you off in particular, it doesn't really make any significant difference to you where/whether it's been advertised - you still have to do the same amount of due diligence.
Third point: it is extremely typical for estate agents to offer you properties that are a small amount above your stated maximum price limit. This is because they know both that people tend to be cautious when saying what they can afford, and that people will get emotionally attached to a property and end up paying more than they originally intended to. Full disclosure, I did this in both properties that I have bought despite knowing this factor up front, so I'm not saying it's some great moral failing to fall into this trap. It's up to you to decide how rigorously you want / need to stick to your originally stated budget or how much flexibility you have, and if you don't have that flexibility then tell the agent that.
Final point: the price of a property can change at any time, up or down, whether it's on Rightmove or not. Until you've signed on the dotted line they are always free to ask for more (and you are free to say no and stick to your original offer) and if they're not getting enough interest in the property then they might drop the asking price. So it could go on Rightmove at any price at all, higher or lower than what you have been told.