Can an altered RSA private key still work as the original?
I would be surprised that a proper system will work with this;

It fails in openssl check.

    openssl rsa -in sample_rsa_prv_mod.key -check RSA key 
    error: n does not equal p q


If you look at an ASN1 parser like ASN.1 JavaScript decoder you will see that your modifications change the modulus. Why the rest didn't corrupt, since the ASN.1 format uses delimiters.

The modulus is not correct. The Khan Academy provides no warning about this i.e. doesn't check as OpenSSL and this implies not a good library. Their implementation may use CRT and from the modulus with $n = p \cdot q$. This is the only plausible explanation since I've tested for some other plaintexts, too.
How the OpenSSL command line behaves with these two private keys;
$cat input.txt
1234567890

$ openssl rsautl -encrypt -pubin -inkey sample_rsa_pub.key -in input.txt

$ openssl rsautl -decrypt -inkey sample_rsa_prv.key -in encrypted.txt
1234567890

$ openssl rsautl -decrypt -inkey sample_rsa_prv_mod.key -in encrypted.txt
RSA operation error
140083785925952:error:0407109F:rsa routines:RSA_padding_check_PKCS1_type_2:pkcs decoding error:../crypto/rsa/rsa_pk1.c:251:
140083785925952:error:04065072:rsa routines:rsa_ossl_private_decrypt:padding check failed:../crypto/rsa/rsa_ossl.c:491:

As we can see, it fails to decrypt. So, the only plausible explanation is again they construct the $n$ from $p$ and $q$. They don't use $n$ in the file.

Note: They use various sources in their JS and probably one fails on their usage.

Textbook RSA little 101

Select target security $t$, today  $t&gt; 2048$
Select public exponent $e$, $e= \{3, 5, 17, 257,65537\}$ are possible candidates;
Generate two unifrom random prime $p$ and $q$ where each has 1024-bit
Form the modulus $n = p \cdot q$
Calculate Euler's totient function $\phi(n) = (p-1)\cdot(q-1)$ ( Actually use Carmichael Function $\lambda(n) = \text{LCM}(p-1,q-1)$ that can produce shorter private exponent $d$)
Check $\gcd(e,\phi(n))=1$, if not 1 return to step 3.
Find $d$ with Ext-GCD such that $e \cdot d = 1 \bmod \phi n$.
Publish the public key as the pair $(n,e)$, the RSA modulus, and the public exponent.
Keep secret the private key $(n,e,d,p,q)$ ( possible more: $n,e,d, p, q, d_p,d_q,d_{inv}$. The values $d_p,d_q,d_{inv}$ are used for CRT based calculation that can speed up modular exponentiation up to 4-time s)

Now encrypt a message $m$ as $c = m^e \bmod n$ and decrypt with $m = c^d \bmod n$.
TextBook RSA as the name suggests should never be used in practice. If you want to use RSA to encrypt some small messages use PKCS#1 v1.5 or OAEP padding to mitigate the attacks, see 20 years of RSA.
If you want to use RSA for signature, use it with PSS padding.
Today RSA mostly used in signatures. For encryption, we prefer hybrid-cryptosystem like key exchange with DHKE (better the ephemeral elliptic curve version ECHKE) and encrypt with authenticated encryption modes like AES-GCM(-SIV) and ChaCha20-Poly1305. It is possible to use RSA-KEM to send the key, too.