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Although it would cost them more time which is why these sort of threats are so pointless. Until people go through with them
This is so true. I learnt the hard way that trying to predict a seller's behaviour by calculating their economic incentive is a waste of time! People do things against their own pure financial interest all the time in buying and selling houses - they get offended, they want to "teach someone a lesson", they don't want someone they don't like living in their precious house etc etc.
If anything the agent should be putting pressure on your solicitors, not you. The seller's solicitor piling pressure on yours is actually them doing their job properly, presuming the other side wants things to move along. Your solicitors should either hurry up (if they are the slow link in the chain and you want things to speed up) or resist that pressure if you don't, because that's what you're paying them to do.
Not actually using your name is obviously unprofessional, if they're annoying you no reason not to call him out on that.
I would be a bit careful with this:
As you're buying the sellers do hold a disproportionate amount of power. If they pull out because they think you're being too slow or you aren't 'motivated' enough as a buyer it won't cost them a penny. Although it would cost them more time which is why these sort of threats are so pointless. Until people go through with them. The reality is you really want this place and not another one you need to keep the seller reasonably happy, and by extension their agent.