Both "to have" and "to help" are verbs - full verbs - as well
Not sure why it is relevant that help is the lexical verb here?
You can see the sentence both as the conditional tense of "have" and also break down the phrase by referring to use of auxiliaries within the tense.
I have read the memoire which refered to palmer - and also looked at a couple of his books - and this is going to be long apologies. Basically as the memoire says, the 60s and 70s were a time when academics were pressurised to be radical - so there are elements to Palmer's work which show his extraordinary knowledge of the English language, elements which were throwing out the rule book (which translated into progressive schools in the UK and somewhat unhelpful EFL methods) and elements where he re-classifies.
In the English verb he indicates that there are 4 tenses - past, present, past conditional and one other - not the future - so I guess you could say this is authority for what you said about the future, but not what you said about there only being 2 tenses - and also authority for what I said upthread about the conditional being a tense. However - these ideas were seen as radical at the time. Also as Palmer said himself, when terms are used in common parlance ie in day to day language, they should (where certain rules are not broken) be absorbed into accepted language. I would say "future tense" is common parlance. It helps neither man nor beast to go around telling people that English does not have a future tense when talking about day to day things. I think one needs to read the work Palmer did in The English Verb about tenses to understand the context.
So, the work Palmer did in The English Verb about why he would distinguish between how the past and present works in comparision with the future is really interesting and his coverage of the myriad of ways in which we can express the future is fantastic, and his work about why tense and time are different concepts (tense does not denote time) and his work on modality in the same book is extraordinary and well worth reading - but - and this is a big but - the work in this book is advanced English - for beginners and intermediates using traditional or formal grammar is going to be a lot more helpful. I found some research papers on the difficulties found by indonesian students in relation to how English is taught as a FL and it looks like his ideas have been taken down into EFL teaching inappriately and many of his ideas have been oversimplified or misunderstood. NB his work in relation to TEFL was aimed at undergraduates or graduates - it is too complex for beginners and intermediates. I think that beginners and intermediates at home and abroad would find it more cogent and easier to learn via traditional grammar which matches how most other countries teach their languages -that is using conjugated English verbs as they were taught prior to 60s and 70s and basic construction of sentences.
It is clear now where some of the ideas in this thread have come from however. The way EFL teaches about modals is so stripped of complexity it is unclear in some places and incorrect in places and this is clear from some of the posts here as just an example.
The memoire gives a flavour of how politicised language had become in the 60s and 70s. It mentions the British Council which I looked up in wiki (not a good source but I just wanted an idea of the history of the organisation) and that is worth reading too as again it gives context and perspective to why things have happened as they have happened.
Palmer wanted to keep some grammar rules but chuck out the rule book in relation to some of the erudite grammar rules (see his book Grammar) - I can see why he came to those conclusions in the environment he worked in but I don't think he realised how many problems would be created with this line of thinking. I think to be critical of some of the erudite rules is fine but to decide they simply do not exist was being as "prescriptive" as the erudite rules he criticised, ironically. Palmer did also explain quite a few of the erudite rules before saying he thought they should disappear which is very helpful as Grammar operates as a text book in some ways. As I said upthread, the educated classes, children who have been to or go to academic private schools, have continued to learn all the rules since the 50s. People who do not know them are going to risk being excluded socially and professionally. Knowing the rules does not mean they have to be used - Churchill famously told a civil servant to bugger awft when the civil servant tried to correct one of his sentences.
There is a place for radical thought and there is a place for basic, correct education. And some of the radical thought of the 60s and 70s had no place in the world at all - see what Simone de Beavoir signed up to in the famous petition about 12 year olds and CSA.
I apologise for the long post. There was a lot there. It was also very interesting so thank you for the source.