What goes in the blank here?
In order for your website to be modern and professional, it is my suggestion that every product ______ more detailed specifications....
My intuition is shall have, but I vaguely remembered that back in high school, my teacher said using past tense is a form of politeness and a sense of softened tone. But should seems to be more forceful, so what should I use here?
As an American English speaker, I would definitely not use shall here. The context where I mainly see shall used are in technical standard documents, where it conveys an absolute requirement, not merely a suggestion.
As for should, in other varieties of English, should might convey a conditional, but in American English it will be understood to indicate a preference, making the suggestion firmer rather than softer.
The example as you presented it seems fine to me.
To soften the language even further, you could use could or might:
...it is my suggestion that every product could have more detailed specifications.
Could conveys a possibility or an option, so it makes the suggestion even weaker. However this softens the suggestion so much that it might appear you lack confidence in it yourself.
As was pointed out in the comments, this is not using the past tense, but the subjunctive, which is used to speak about possibilities rather than known facts. In English the past and subjunctive forms of can are the same: could. similarly, the past and subjunctive forms of shall are both should.
You seem to have realized that suggest that something ᴠᴇʀʙ doesn’t take a normal third-person singular inflection. Good for you.
Suggest is a verb that takes a stuffy subordinate clause, with a “modally marked” verb, meaning either a bare infinitive or one prefixed by an actual modal. This is what used to be the old present subjunctive in English and related languages, but we use the bare infinitive for it now because there’s no special inflection for it any longer.
Your original with have is perfectly grammatical, although I might shorten it up a bit:
For your website to be modern and professional, I suggest that every product have detailed specifications.
That’s pretty stiff and formal. Your second choice, should have, is probably even stuffier, but you can use its should to come up with something simpler and more direct:
For your website to be modern and professional, every product should have detailed specifications.
Or even:
Every product needs detailed specifications for your website to be modern and professional.
(That’s my own preferred formulation here.)
Your final version with shall doesn’t really sound right. This could be appropriate in legalese where shall carries the weight of must, but that doesn’t make sense for a suggestion, only for a demand or order or requirement.
