Possession after one tenant gives notice to end a joint tenancy

Possession - Section 8 and Section 21

My question is based on recent experience in trying to gain legal advise from two different solicitors.

I let a property to two joint tenants, one of whom has left and has given me written notice to quit, I received the notice on the 23rd July. The rent due date is the 15th of every month and therefore the notice requests that the tenancy ends on the 14th September in order to give one full months notice from a tenancy period. Because the tenancy is a joint one, I understand that the remaining tenant also needs to vacate the property by the 14th September unless a new contract is agreed.
I also understand that if the remaining tenant stays in the property after the the 14th September without a new contract being agreed, she will effectively be living there illegally at which point I can apply for a possession order using N5 and N119 forms along with a copy of the notice.

The N5 and N119 forms are a little complicated to someone not used to filling them in so in anticipation of a problem I asked two separate solicitors if they could help if required. Both told me that if the remaining tenant remains in the property I would have to serve a Section 21 even though the law appears to state otherwise. The written reply I received by one of the Solicitors is as follows, I have deleted the names of the tenants:

"Although one tenant has vacated the property and provided you with a document ending the tenancy I would need to consider that point further to ensure that the tenancy of xxxxxx has indeed been brought to an end. My costs for considering that point will be in the region of £xxx plus vat.
Should xxxxxxx not vacate the property then the first step will be to issue either a section 21 Notice if there are no arrears, or a section 8 notice if there are arrears. It is in some cases appropriate to issue both notices. You would not need to wait until the 14 September 2021 to issue that notice if you do want to gain possession of the property. A section 21 notice currently provides the tenant with 4 months to vacate"

The reply I received from the other solicitor was verbal but along similar lines i.e. that I could not rely on the tenant's notice to gain possession even if it is correct and legal. Whilst I totally believe that the advice given to me by the NRLA and Shelter is 100% correct i.e. that a possession order can be applied for once notice from one of the tenants expires, I am concerned as to why Solicitors are advocating a different route i.e. Section 21 or 8?
I am keen to know if anyone has had to apply for possession after a joint tenancy has been ended by one of the tenants and what route they took to do so.