where can a Omani expat travel to without applying for a visa?

No such thing as visa free access for GCC expat in GCC or outside countries; usually it is visa on arrival and that too in few countries.  I have listed those in point 3 but it does vary from time to time based on nationality / country of residence in GCC.

What you get is:

1) Visa on arrival in GCC countries  on the basis of your Oman residency and pay for the visa.  You can get these on arrival in Bahrain, Qatar (still for some gcc countries) and Kuwait while for UAE, you apply online.  For Saudi, it is a full visa process
2) Ability to apply for other countries visa from Oman as you are a resident.  In some cases, it is an easier and quicker process then if you were to apply from Pakistan
3) Jordan, Georgia, Tunisia and Armenia allow this but check as it varies i.e. sometimes only Indians or Saudi iqamas or some nationalities barred despite possessing GCC residency etc.  Also expiry date of your residency is also a factor; ideally should be valid for 6 months or more when you visit

As an expat in Oman , your status - as far as visa application is concerned, would not change.

The requirement of Visa would be based on the passport that you hold.
Of course the time required for processing may change ( for example US visa - taken from US Emabassy in Oman was just a 30 minute affair - they of course would ask you to collect it after 3 days. The same thing in India would take lot more time (based on availibility of interview slots - a pure demand supply mechanism).

Coming to your earlier and main question - its the strength of your passport which decides whether you require a visa to visit a particular country or not.

For example, as things stand now - Singapore would be the country with most "powerful" passport - and if you happen to hold one - you do not need any visa to travel to 127 countries . That same number for Pakistan (Visa free)  would be 34.

The following website would give you a ready reckoner .

https://www.passportindex.org/byRank.php

However, it is always advisable to check with the specific embassy website as the site may not be updated real time.

Thanks