Traveling to the US with a visa, but not for the purpose of the visa - allowed?

Maybe...

Although I have an E2 and enter the US regularly, I applied for an ESTA for a holiday and entered the US on it. I was questioned and berated for entering the US on an ESTA and told I should have entered on the E2. I stated that I wasn't entering for work, but for a two week vacation with my family who were travelling with me. The CBP agent asked his supervisor who also seemed unsure whether I should have entered on the E2 or the ESTA. In the end I was stamped in on the ESTA but he wasn't happy.

On a subsequent E2 entry, I asked the agent what I should do if I enter for vacation and he said I could enter on an E2. My reading is the same as the other answer and that you should be entering on an ESTA but it almost seems that they consider you as someone who's left the US while on the E2 rather than someone entering on it.

The main reason I got the ESTA was to avoid not being able to do online check-in due to visa checks, which has gone away since getting the ESTA. The airline assumes I'm using my ESTA so lets me get my boarding pass and I use the visa on arrival.

My advice would be to get the ESTA and enter on it, but don't be surprised if you meet an agent who is unhappy about it.

NO. According to the USCIS Requirements for E-2 Visa:

Requirement 5 - You must be coming to the United States to develop and direct the enterprise.

You clearly are not coming to do so. A border immigration officer will be right to deny you entry and revoke/cancel your visa for coming for a reason other than that for which the visa was issued.

You can come on VWP if you're eligible and need to apply for an ESTA if coming by air or water. Apply at least three days before travel

Best.

I'll answer an unequivocal "yes": you can vacation in the US while in E-2 status.

To clarify, E-2 status imposes no particular requirements or restrictions on time spent in the US. You are free to pack up your household and stay in the US for the duration of the status (and E-2 dependents can apply for EAD), but you can also freely travel and spend no specific amount of time in the US. See, e.g. 9 FAM 402.9-4(C). The restrictions on time in the US you are observing hence have little to do with your E-2 status, and are much more likely aimed at maintaining non-resident tax status in the US (in which case the number a conservative tax lawyer would be telling you is probably around 120 days). Nonimmigrants with work permission will have shown their qualification for that status based on the work they are doing but certainly do not have to spend 100% of their time in the US in that status doing that work; they may also take vacations and do most other activities generally allowed to visitors. From the point of view of US Immigration an E-2 is "travelling" when he is outside the US, an E-2 is free to travel without restrictions on frequency or duration and the fact that you might plan to travel immediately before and after a vacation in the US just isn't very relevant to anyone.

Note that an ESTA isn't really a "status" in the US, it is just a predetermination that you are likely to be admissible there that gets you on a plane flying to the US. Your visa will also get you on the plane, but if an ESTA makes getting on the plane easier feel free to buy one. Once you get to the US, however, you are free to request entry in any non-immigrant status you qualify for (or even several at once if there's a reason for that). That said, if you have an E-2 visa then, as the other answer indicates, the immigration officer is going to find it odd if you request a WT admission instead since there is nothing someone in WT status can do that an E-2 can't do, but a WT entry forecloses the possibility of changing your mind and doing some work on that entry. Not entering as an E-2 is going to be swimming upstream, so it is likely to be much less hassle just to ask for readmission as an E-2.

The status of your entry will have no effect on the count of days you are keeping. For tax purposes all days in the US in any status (and no status) are counted, so you'll be counting vacation days in your total no matter what.