This post has sparked lots of emotion, judgement (good or bad) and controversy and I dont know why.
You mention you aren't too familiar with estate agents and are asking for advice on if their communication or actions are normal and or following procedure and or ethical.
Simple answer: YES.
From the timeline, your first enquiry was after the advert had been live for 2 weeks. You say the house is a bargain price on a desirable road where done up properties go for 2mil+. Therefore, it is not unusual to expect that within that 2 week period there was a great deal of interest from buyers. It sounds like during that period before your initial enquiry, they had viewings and presented an offer which the vendor has accepted.
Typically although not always, people make offers contingent on no further viewings or offers.
I see your frustration and confusion on the basis that the house is clearly still listed as for sale on rightmove. This implies that other potential buyers should still have open access to view and make offers.
Unfortunately that is just not the case UNLESS the vendor has accepted the offer in principle and told the agent (and their buyer) they reserve the right to continue with viewings and listen to other offers. If they accepted the offer on the premise of no further viewings and gave those instructions to the agent then the agent is simply responding to enquiries under the instructions of their client - ie decline further viewings.
My property purchase was done with 2 different agents representing buyer and vendor on both ends....our properties (london) remained on rightmove until contracts were exchanged and only then did they become marked as sold.
Some buyers insist on the advert being updated to say under offer as a bare minimum once their offer is accepted but I dont believe that any agents are obliged to do so. It will never say SSTC until much further down the conveyancing process.
I had a similar experience when I was searching and made multiple calls and enquiries to agents to request viewings...asked why on earth it is still listed as for sale weeks or months down the line if an offer was already accepted. As people have said, it is a common marketing strategy and whilst irritating, not unusual at all. I asked to view a property that I loved but the agent stood firm and said no, the vendor has accepted an offer and wants to proceed so regardless of a new buyers position, ethically they were not willing to show the property... in honesty, I think thats completely fair IF someone's offer stipulates no further viewings.
As others stated, there is nothing at all stopping you from making a formal written offer via the estate agents putting forward your position as proceedable cash buyers. IF the vendors have not explicitly said do not show us further offers then yes, by law the agents must present it. So you have nothing to lose.
But the agent is absolutely in their right to decline interested parties to view once an offer has been accepted. It is therefore up to you to decide if you want to write a formal offer without viewing the property in person.
In my view (and I dont profess to be a property law expert or an estate agent expert), it really is quite simple.
Incredibly difficult to take the emotion or frustrations out of house buying especially in this hideous market but I think this post answers your original question objectively and simply.
Best of luck whatever you decide to do and I hope whatever the outcome is, you find a home that you love.