The 181 is a "last resort" measure. This is why they rarely exist.
Petition is just appeal for non-appealable issues.
But petitions are much, much rarer than appeals. I probably appeal about 10% of the cases that I work on. I have hardly petitioned at all - mostly because firms don't believe they are effective and clients will not pay for the service fees.
How many appeals have you had, JAE? If it's more than zero, the you've had more appeals than petitions. Why should that be? Are you better at entering amendments, making objections, making restriction requirements, and making actions final, than you are at rejecting claims? Or does appeal simply work better than petitions do at correcting examiner errors?
Take restrictions: If the examiner makes a questionable restriction, applicant should traverse the restriction in their reply. If the examiner still upholds the restriction and makes it final, they have to explain why in writing. If applicant still disagrees they can call the examiner and ask for an oral explanation. It is extremely rare to go past this point, which would require both the examiner saying "No, I'm right", and applicant saying "No, I'm right". One side usually concedes by this point.
1. I have never had an examiner admit that a restriction requirement was wrong. I have had a SPE tell me point blank that a flagrantly wrong classification, which required "cryptography", and said that "mere nominal recitation of cryptography is not enough to meet this classification", was correct, even though the independent claim and dependent claims had zero reference to cryptography (not even nominal cryptography). The SPE told me I could petition. The firm and client would not authorize it.
2. There is no requirement to talk to the examiner on the phone before petitioning.
Petition is just like appeal for non-appeal issues. It is a "last resort" in that sense, but that doesn't mean that examiners don't force applicants to use their option of last resort. I appeal all the time, and I win 90% of my appeals. But I usually don't have authorization to file petitions, and that's not uncommon - it's the same reason why you never see them.
The same scenario exists for finality arguments. You present it in writing and/or a phone call with the examiner. Maybe even pull in their SPE. Very rarely does this result in both sides still butting heads against one another.
You have more experience with other patent attorneys than I do. I can only give you my own experience at my firm. There are lots of applicants, and foreign assignees, who don't like to appeal anything. Either they focus on the BPAI affirmance rate, without considering that most appeals don't get to the BPAI, or they don't care about their claim scope enough to justify the delay and expense of appeal. But, at least at my firm, and at many others, it's very common to appeal. JimIvey has said that he has a high appeal rate, too. I know that many other practitioners do. In many cases, it's the only way to get the examiner to withdraw bad rejections.