When we use "would" to express uncertainty, do we use past tense in the noun clause?

Question

I came across the use of would for expressing uncertainty. (Article: https://www.englishclub.com/grammar/verbs-modal-would.htm)

Examples from the article:

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would for uncertainty

(1) He would seem to be getting better. (less certain than: He seems to be getting better.)

(2) It would appear that I was wrong. (less certain than: It appears that I was wrong.)

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According to the examples from the article, does that mean that when using would to express the uncertainty, we can treat it like the would doesn't exist and the main verb is just a present-tense word, but just with less certainty?

So when we use "would" that way, do we use past tense in the noun clause?

Which is correct in the examples below? Present tense or past tense?

Do (3) and (4) mean the same?

Do (5) and (6) mean the same?

(3) He bought a new game yesterday. He would say he wants/wanted to stay at home and play the game today.

(4) He bought a new game yesterday. He might say he wants/wanted to stay at home and play the game today.

(5) He is quite stubborn. He would insist that he is/was not guilty.

(6) He is quite stubborn. He might insist that he is/was not guilty.

Answer 1

Your questions about tense:

In all of the examples you provided you are using reporting verbs. Remember that reporting verbs aren't only "said" or "think" they can be any verbs which help relate an idea expressed elsewhere. Here's a list of reporting verbs.

When we write something in reported speech what follows can be in the present or in the past. That depends on whether you're expressing an idea that is generally true or the idea was true at some point in the past. Here's an explanation of when to use the past or present in reported speech.

Using would for uncertainty:

In the two examples in the article you cited, they use the verbs "seem" and "appear" which are two verbs that are used for uncertain statements, or in other words, speculation. In your examples, you are using reporting verbs which aren't used for making uncertain statements.

You can only use verbs that are speculative (uncertain) using would in this way. Here they are:

Look, feel, seem, appear, sounds, and not many others and these.

Your examples re-written:

He bought a new game yesterday and he isn't answering his phone. It would appear that he wants to stay home and play it all day.

The prisoner insists that he is not guilty despite the evidence. It would seem that he has convinced himself of the idea as a way of dealing with his predicament.

You say that you are detail oriented and self-motivated. It sounds like you would be a great fit for our team. We'll give you a call when all the interviews have been completed.

Answer 2

The answer by @yuri-teacher-english is excellent, but just to clarify something in terms that might help a learner of English:

The verb conveying uncertainty in the sentence does not affect the time when a related event takes place.

Consider the following:

He might claim that he was completely innocent in 2010.

He might claim that he is innocent now, but admit he was guilty in 2010.

He might claim that he will be found innocent when the case comes to court in 2022.

All are equally valid, but all say different things about when Fred is likely to claim he is to be considered innocent.

The time when Fred makes his claim is the same in each sentence - an indefinite moment that could exist now or in the future. What changes is when Fred wants his innocence to be considered to exist.

Answer 3

In these contexts, "seem" and "appear" are the root verbs, which "would" is modifying.

"Seem" implies uncertainty. It directly means "appears to be" instead of "is". "Appear" is similar, it also says something is presenting in a way that it might not me.