Is there a liability limit for a carry-on bag that was forcibly checked plane-side?

As an example, read American Airlines Conditions of Carriage, specifically Carry on Baggage and Liability.

The suitability of carry-on baggage will be determined by American. Each customer will be limited to one carry-on bag and one personal item. A personal item includes a purse, briefcase, laptop, or small book bag style backpack. A personal item must be smaller than your carry-on bag and must fit under the seat in front of you. Fragile or valuable items, such as keys, medication or computers should be carried in your personal item.

That’s the first point. A personal item wouldn’t be checked, since it is supposed to fit under the seat, not in the overhead, and all valuable items should be in it, and not carry-on baggage.

The second point is under liability:

American does not assume liability for any of the following items in or as checked baggage: antiques, artifacts, artwork, books and documents, china, computers and other electronic equipment, computer software, fragile items (including child/infant restraint devices such as strollers and car seats), eyeglasses, prescription sunglasses, non-prescription sunglasses and all other eyewear and eye/vision devices whether lenses are glass, plastic, or some other material, furs, heirlooms, keys, liquids, medicines, money, orthotics, surgical supports, perishable items, photographic, video and optical equipment, precious metals, stones or jewelry, securities and negotiable papers, silverware, samples, unique or irreplaceable items or any other similar valuable items.

If your carry-on is checked, and it contains valuables, they are not liable. Period. Valuables belong in your personal item.

Next paragraph starts out:

With the exception of strollers, American does not accept these items in or as checked baggage and assumes no responsibility or liability for such items, regardless of whether American knew or should have known of the presence of such items in checked or transferred baggage.

And then, the piece de la resistance is paragraph 6 under liability:

American assumes no liability for articles carried in the passenger cabin.

It is my assumption, that every other airline, large or small, will have quite similar statements as these in their fine print.

The bottom line: mail, and insure.

The airlines all make it clear that they do not guarantee space on board for all items that are eligible carry-on luggage, as pointed out in another answer. I would presume that you cannot claim a higher loss allowance for a bag just because it is forced to be checked during boarding. I wouldn’t rely on an agent’s word at the gate, either.

The best way to protect yourself is to plan to remove the valuable items and carry them on (often times there is enough space for individual items even if there is not enough space for a standard carry-on suitcase).

There are certain types of smaller aircraft that do not accommodate standard carry-on suitcases. Check your itinerary to ensure you aren’t flying on one of those planes. They include Canadair Regional Jets, many Embraer regional jets and all prop style planes like the Dash-8 or ATR. Larger bags on these planes are always checked at boarding and returned to you upon arrival at the gate.

If you cannot remove the items and carry them on safely, then you should be prepared to make the case with an airline agent to be moved to another flight. I don’t know how successful you will be, and you may have to wait out many full flights before it happens, but that may be your best best to avoid checking a larger but valuable item.

Edit: I will mention something from a helpful comment by Tor-Einar Jarnbjo: you might be able to purchase a second seat to accommodate a very valuable item like a musical instrument. Contact your airline to discuss this option.

Without checking every airline’s conditions, most (if not all) airlines only allow carry-on luggage under the provision that there is available space. Basically, there is usually not enough stowage in the cabin should all passengers bring the maximum allowed amount of carry-on luggage, so it is bound to happen now and then that some intended carry-on luggage will have to be moved to the luggage room.

I find it odd that noone at US Airways were able or willing to answer your question regarding this issue. The relevant part in their contract of carriage is pretty clear:

Carry-on baggage allowance may be restricted due to lack of space on
board the aircraft.

They are not even trying to hide the fact behind unintelligible legalese, but stating clearly that they don’t guarantee you any right to bring carry-on luggage on any flight.

In Delta’s terms and conditions, you will find the following disclaimer:

Acceptance of carry-on baggage is subject to space availability on the
aircraft at the time the passenger boards. If adequate space is not
available, Delta may require that the baggage be checked.

While United writes in their contract of carriage:

Operations, space constraints, security directives and/or other safety
considerations may require limitations to the allowable Carry-on
Baggage on a specific flight.

That covers the three airlines you were especially asking about. Other airlines are likely to have similar disclaimers in their terms and conditions or contract of carriage.