The question is usually about the whole domain, not one record
When something is wrong with a domain, you rarely know which record is responsible. Mail is bouncing - that could be the MX records, or an SPF record in TXT, or a DKIM selector, or the domain resolving somewhere unexpected. The site is not loading - that could be A, AAAA, a CNAME pointing at something stale, or nameservers that were never updated at the registrar.
Looking at all of them together is how you find the one that is wrong, and it is usually obvious the moment they are side by side: a CNAME still pointing at the old host, an MX record for a provider you stopped using, two SPF records where there must only ever be one. Checking them individually means knowing the answer before you start.
What a lookup actually tells you, and what it does not
A public resolver answers from its cache. What you get back is what that resolver currently believes, which after a change may be the old value and may be the new one depending on how much of the previous time-to-live is left. That is why one person sees the new site and another sees the old one, and why both are correct.
So a lookup confirms what the public internet is being told right now. It does not confirm that your change is wrong, and it does not tell you how long the old answer will persist elsewhere - that is governed by the record time-to-live as it was before you changed it. Lowering the TTL a day before a migration is the standard trick, and it only works if you remember in advance.
TXT records are where the surprises live
TXT has become the place everything gets bolted on: SPF, DKIM selectors, DMARC policy, domain verification strings for a dozen services, and whatever a previous administrator added and never removed. It is also the record type with the most footguns, because several of the things stored there have rules about how many of them may exist.
The classic failure is two SPF records on one domain, which is invalid - receivers are entitled to treat it as a permanent error, and mail that was being delivered stops. It usually happens when a second service is added and its instructions are followed literally rather than merged into the existing record. Seeing every TXT record at once is how that gets spotted.