As far as I know, if there is a vowel before the final /t/ or if there is the letter /n/ before the final /t/, we can make an unaspirated /t/ sound in American English. For example: "hot", "cut", "cent", "can't", "cat", "pit", "sit", "eight" etc. Can we do the same thing if there are other consonants than /n/ before the final /t/ as well? Or do I have to aspirate the /t/ in the words like "best", "worst", "greatest", "first", "belt", "port", "part" etc.? I mean, I know that when this kind of words are in the middle of a sentence, we can make the unaspirated /t/ sound. But can I make the same unaspirated sound when those kinds of words are at the end of sentences or clauses, or when they are the only word in a sentence? For example If I say "I am the best player in the team", I know that the /t/ in "best" can be unaspirated. But can it be unaspirated also in a sentence like "I am the best.", where the /t/ is the last sound of the sentence and there is another consonant before it? If the sentence was "I am so hot", the /t/ could be unaspirated. But as I said, I am not sure about the situations where the clause or the sentence ends with /st/, /lt/, /rt/ etc. I need help on this.
I'm not a trained linguist, just a very observant student of language who pays attention to how my compatriots speak. I would say yes, you can make the unaspirated /t/ sound at the end of all of those words and sound like you've been speaking American English all of your life, whether they're in a sentence or by themselves.
I watched the video you mentioned. I disagree that speakers of American English always pronounce the final stop without aspiration. Sometimes they do and sometimes they don't. In fact I don't think you can say that all speakers of American English treat any final stop in any situation the same all the time. There are many variations.
It's not true that you won't sound native if you pronounce a final stop with aspiration. When Americans are speaking slowly or emphasizing what they're saying the will sometimes aspirate the stops that otherwise wouldn't be aspirated.
Another variation is not to pronounce the final consonant, particularly when speaking quickly and casually. For instance,
'That's the best thing'
might often sound like
'thas the bes thing'.
This is the same, for practical purposes, as your example of 'old' and 'gold' without the aspiration.
Regardless of what the video says, some speakers make glottal stops in place of the final /t/, to a greater or lesser extent.
So, yes, you can feel safe in making unaspirated stops in all the situations you mentioned in your question. However, I'd suggest listening to how the people around you are speaking and mimicking them. You'll find there are subtle variations that will make all the difference in how you sound.
The meaning of the term "aspirated"
Warning: I have no phonetic training, so this is just pieced together based on a few documents that I have read.
In IPA, the symbol /ʰ/ is used to mark aspiration, and this seems not to be a coincidence. Apparently, aspirated stops are followed by a noise resembling the sound /h/, in addition to the release noise of the stop (Kiss).
One common way of measuring aspiration is to look at the "voice onset time": the time between the release of a stop and the start of voicing. Aspiration is associated with a higher voice onset time, but there is a gradient between aspirated and unaspirated stops. Obviously, voice onset time can only be measured when the stop is followed by a voiced segment.
Daniel Harbour's answer to the ELU question "Can a plosive be pronounced without an audible release sound following it?" says:
(1) The stops t, p, k, when syllable-final, undergo glottal reinforcement in English. This minor glottal occlusion does not wholly impede the airstream. So, when the stop is released, the remainder of the air is released too. This is reminiscent of an unvoiced schwa, which accounts for what you hear after the t in cat. In terms of IPA transcription, one tends not to write the fine, automatic phonetic detail, and, so, for English, one marks only the preglottalization, as in [kʰæˀt].
(2) Crosslinguistically, the behaviour of syllable-final t, p, k varies. In Kiowa, for instance, as described by JP Harrington, t and p undergo complete glottal closure and are unreleased. In German, they are lightly post-aspirated. There are, however, only a certain number of perceptually distinct things you can do with your articulatory tract. So, my guess is, there'll be other languages that behave as English does.
As far as I can tell, word-final voiceless stops only seem to be described as phonetically "aspirated" when they are followed by an audible [h]-like sound that is sustained for a relatively long time. It doesn't seem to be standard to use the term "aspirated" to refer to all word-final voiceless stops that have audible release of any kind. I would guess that, as with stops in word-initial position, the classification would be gradient: stops that have no noticeable "puff of air" sound would be classified as having "no audible release" (also, perhaps misleadingly, called "unreleased"), stops with audible release that have a very small "puff" of air would be classified as unaspirated with audible release, and stops with audible release that have a "puff" or air of comparable duration to the VOT of a word-initial stop would be classifed as aspirated.
When talking about English phonology, the term "aspirated" is usually only used to describe a set of allophones of the voiceless stop phonemes /p t k/ and the affricate /tʃ/. (Some other languages such as Hindi are sometimes said to have "aspirated voiced stops"; in that context, it refers to breathy-voiced stops).
I suppose, since English word-final voiced stop phonemes may be phonetically voiceless, they might be able to have some phonetic aspiration when they are audibly released, but I don't think that is all that common. No phonological description of English that I have read mentions any aspirated allophones of voiced stop phonemes.
The use of aspirated allophones in English
As the notes on that video say, "At the end of most words (and syllables) stop sounds are pronounced without a puff of air. For example, "lap," "sheep," "update." [...] if you make a puff of air, you’ll be understood, but you won’t be speaking with an American accent." There is no exception for words that end in consonant clusters. It is usual to not aspirate the stop.
In fact, I think a speaker would probably be less, not more likely to aspirate a word-final stop after /s/, because stops are typically not aspirated after fricatives in the syllable onset (e.g. star, spoon, skin), so I think English speakers have a general preference not to aspirate a stop that comes after a fricative in the same syllable. (See Kiss.)
I would guess words with a liquid followed by a word-final stop like "belt", "port", "part" are probably about as likely to be pronounced with aspiration as words with a vowel followed by a stop. Note that in American English accents, /r/ typically does not prevent a following /t/ from becoming voiced and flapped/tapped before a vowel-initial word--e.g the /t/ in "art" would be voiced and flapped/tapped in the context "the art of...".
The use of audible release in American English
I don't think it sounds bad to have a weak audible release for uttereance-final plosives in American English.