To avoid opinion-based answers, this type of question requires an answer based on an authorative reference with credible and/or official sources.
The following excerpts are quotes from W3C Validator Help & FAQ that addresses Why are browsers accepting invalid HTML in the first place?
and some other demonstrated concerns related to that.
About Markup
Most pages on the World Wide Web are written in computer languages
(such as HTML) that allow Web authors to structure text, add
multimedia content, and specify what appearance, or style, the result
should have.
As for every language, these have their own grammar, vocabulary and
syntax, and every document written with these computer languages are
supposed to follow these rules. The (X)HTML languages, for all
versions up to XHTML 1.1, are using machine-readable grammars called
DTDs, a mechanism inherited from SGML.
However, Just as texts in a natural language can include spelling or
grammar errors, documents using Markup languages may (for various
reasons) not be following these rules.
[...]
Concepts
One of the important maxims of computer programming is: "Be
conservative in what you produce; be liberal in what you accept."
Browsers follow the second half of this maxim by accepting Web pages
and trying to display them even if they're not legal HTML. Usually
this means that the browser will try to make educated guesses about
what you probably meant. The problem is that different browsers (or
even different versions of the same browser) will make different
guesses about the same illegal construct; worse, if your HTML is
really pathological, the browser could get hopelessly confused and
produce a mangled mess, or even crash.
That's why you want to follow the first half of the maxim by making
sure your pages are legal HTML.
[...]
Validity might not mean quality, and invalidity might not mean poor quality
A valid Web page is not necessarily a good web page, but an invalid
Web page has little chance of being a good web page.
For that reason, the fact that the W3C Markup Validator says that one
page passes validation does not mean that W3C assesses that it is a
good page. It only means that a tool (not necessarily without flaws)
has found the page to comply with a specific set of rules. No more, no
less. This is also why the "valid ..." icons should never be
considered as a "W3C seal of quality".
Unexpected browser behavior might mean that they actually don't accept invalid markup
While contemporary Web browsers do an increasingly good job of parsing
even the worst HTML “tag soup”, some errors are not always caught
gracefully. Very often, different software on different platforms will
not handle errors in a similar fashion, making it extremely difficult
to apply style or layout consistently.
Using standard, interoperable markup and stylesheets, on the other
hand, offers a much greater chance of having one's page handled
consistently across platforms and user-agents.
[...]
Compatibility problems
Checking that a page “displays fine” in several contemporary browsers
may be a reasonable insurance that the page will “work” today, but it
does not guarantee that it will work tomorrow.
In the past, many authors who relied on the quirks of Netscape 1.1
suddenly found their pages appeared totally blank in Netscape 2.0.
Whilst Internet Explorer initially set out to be bug-compatible with
Netscape, it too has moved towards standards compliance in later
releases.
[...]
Relying too much on 3rd party tools
The answer to this one is that markup languages are no more than data
formats. So a website doesn't look like anything at all! It only takes
on a visual appearance when it is presented by your browser.
In practice, different browsers can and do display the same page very
differently. This is deliberate, and doesn't imply any kind of browser
bug. A term sometimes used for this is WYSINWOG - What You See Is Not
What Others Get (unless by coincidence). It is indeed one of the
principal strengths of the web, that (for example) a visually impaired
user can select very large print or text-to-speech without a publisher
having to go to the trouble and expense of preparing a separate
edition.