Do you want to choose which solicitor you use? If you do, then getting the estate agent to arrange the HIP will often deny you that choice.
The estate agent will arrange the HIP either through a particular firm of solicitors who will generally only do the HIP if they get the conveyancing, or through a HIP Provider who will "introduce" you to one of their panel solicitors who will probably be in a distant location. That panel solicitor will pay a "referral fee" to the HIP provider. To pay that fee the solicitor has to charge you more. Do you want that?
If you choose your solicitor before you get involved with estate agents then the solicitor can arrange the HIP. If you pay him upfront then the HIP is yours and you can change solicitors later if you want. Will that apply with estate agents?
Of course at the moment all of this is only going to apply to 4 bedroom houses from 1st August so it will be much less important in the short term than it was first thought.
Local Authority Searches used only to be done directly through the Council in quesiton and solicitors had to pay whatever the Council charged, which varies from about £70 (I'm not sure where this is) up to about £280, with an average outside London of about £150. These searches can now be done through large private search companies such as STL, OneSearch, Richards Gray. These companies keep databases of much of the information but have to send their agents into the Council offices to retrieve some of the stuff. Generally they are cheaper, and usually charge a standard fee of about £100 for a search anywhere in England and Wales.
If the buyer is getting a mortgage then usually the buyer's solicitor will also act for the mortgage lender and will have to follow the rules set out by it. Some lenders will not accept these search agency searches, e.g. HSBC and Britannia Building Society. Most either say "Yes" they will accept them or "No, but you may do so at your own risk". In the latter case in the unlikely eventuality of something going wrong and it turning out that an adverse matter existed that should have been shown on the search, but wasn't, then if the lender repossesed and lost out on a sale, it would sue the solicitor (who had taken the risk). The solicitor would then sue the search agency, who would recover from its own indemnity insurance.
Personally, I think the risk is acceptable, and therefore at the moment when acting for buyers (unless I have a lender that will not accept search agency searches at all), I use a search company.
Some solicitors won't take that risk, and so would say that they wanted the proper local search from the Council. Many of the large HIPS Providers will be streamlining their service by using private search agencies, some of whom have made commercial arrangements with HIPS providers. The result is that if I am providing a HIP in the future I would normally advise that the proper local search is carried out to avoid arguments from buyers' solicitors. This could cause delays if they then insist on carrying out another search, not to mention an annoyed buyer who thought he didn't have to pay for the search!
Therefore I think what easylayers.co.uk say is about the searches is broadly right, although I am not sure that people are generally best off using a large online set up because they can be remote and impersonal.
As a conveyancing solicitor I believe the information given in the post to be useful but I accept no liability except to fee-paying clients.