Considering the report of an exposed live wire by the meter change "engineer" then it seems very unlikely it had a current EICR (or if it did, not one done by a competent person).
I would start by giving the landlord the benefit of the doubt and asking, in a civil but very insistent manner, that a EICR is now conducted by an registered electrician for the whole house and that you are given a copy of it (and full details of who does it) for peace of mind.
Having a lot of sockets is not a bad thing in itself, it is better than having dodgy extension leads everywhere, but as an ECIR typically does not check every socket (usually a sample of them) you should draw attention to your concerns. In particular, if there are any that don't look or feel right (cracked, evidence of overheating, imprecise switch action, unreliable connection, etc) then you should identify them (e.g. put on a sticky label) and ask the inspection specifically looks at those and they are replaced if there is any doubt.
If the landlord is not cooperative then take it further with legal advice, but it might be they have had a nasty surprise as well and are now willing to sort it out properly by getting a professional in to do what is needed.
[automerge]1589184752[/automerge]
While there should be an EICR now due to the accident, if you wanted to do independent 100% testing of the sockets you can buy basic socket testers for around £10 and they will pick out the most serious of faults, but they do not show a poor earth connection, only a complete loss of it. For around £50 you get better ones that are able to analyse the earth impedance, such has:
https://www.NoLinkingToThis/p/kewtech-loopcheck-107-advanced-plug-in-socket-tester/4670j
But remember testing is
secondary to inspection: if a socket looks or feels bad it should be replaced, otherwise it should then be tested and if it measures bad its an electrician's job to investigate further (replace and check wiring, etc).