Buyer's solicitor "forgot" to order the searches

[Bigbangtheory]

13 weeks after accepting the offer on our house, exchange imminent, date set for completion, when out of nowhere, our buyer's dropped the bombshell, oops, we haven't ordered our searches!" But don't worry, we will pay extra for the express service and they will be back in 48 hours."
So OK, 48 hours isn't too bad....
Following day, I am told that the searches won't be back until after Easter, as whilst the Environmental Search and Drainage Search usually take a day or two, the Local Authority Search is a search carried out by a search provider with the Local Authority’s records and the turn around for this is 2-3 weeks.
I find it unbelievable that we have got this far, only to be told that my buyer hasn't had their searches done! My solicitor has suggested our buyer takes out a "no search" indemnity policy, but so far, there has been no response.
We have been ready to exchange on our onward purchase since the end of January.
I am beyond frustrated!
Has anyone else had this happen? My estate agent said it is the first time he has had it happen, in over 20 years.
There is nothing I can do but suck it up, sat here surrounded by packing boxes.
There must be a better way to buy and sell houses than this archaic system!
Just needed to vent!

You would think Solicitors have a check list wouldn’t you. When we sold my father in laws house, all was going well until our Solicitor found an error in the title deeds so we had to have them put in our names before the sale could go through, we were mortified as we told father in law to make the changes. It delayed the sale 2 weeks and then the buyers Solicitor wanted everything signed electronically, but my Solicitor was old school and wanted everything signed in person, so another delay. I believe in Scotland house sales are completed and exchanged within 2 weeks, if they can do it, why can’t we do it in England.

@Mindymomo We have a deeds issue too. House has been going through almost a year, finally got to the end and the buyer's solicitor has found an issue with the deeds which now need to be retrieved from a repository somewhere 😡.
Absolute joke.