I'm buying a bus ticket to get to Boston 23 hours before my flight to ensure I reach my departure airport on time.
If I didn't arrange bus transport well in advance, what legal remedies would I have if a delayed bus caused me to miss my flight?
There are no remedies. Greyhound's terms and conditions explicitly disclaim liability for delays:
In no event shall carrier be liable for consequential or incidental damages for loss, damage or delay, including weather delays.
And as far as I know, there are no laws in the US that would require them to give you any kind of compensation beyond what the contract specifies (i.e. none).
Maybe you can find travel insurance that would help cover your losses in case of such a delay. (Read the policy carefully to make sure it would cover such an event.) But Greyhound won't. By default, this risk is assumed by you and nobody else.
If you book separate segments from separate companies, they don't owe you anything for a missed connection. They got you to the agreed location, their job is done.
If you book a multi segment ticket "thru" from a single airline, then that airline is responsible for your missed connections. So for instance if Delta sold you Glenwood Springs to Dubai, with GS to Denver and JFK-Dubai being codeshares with Amtrak and Emirates, and the Amtrak train is super late... It's all on Delta to get you to Dubai.
If you booked the Amtrak segment on your own (to save money), then tough beans.
So that thing you are looking for, happens with thru ticketing. Buy tickets that way. If you can't, buy missed connection insurance.
You cannot coerce any company into taking that liability. It lies with you. Regardless of the form of transit, when you book separate tickets from point A to point B and point B to point C, the liability of a missed connection at B lies solely with you.
If you want someone else to be liable for getting you from point A to point C, you need to purchase that on a single ticket or else purchase missed connection insurance, wherein you pay an insurance company to assume that liability.
One possible course of action in this situation is to make alternate arrangements once it becomes clear that you're not going to make your flight, and then present a sympathetic story to the company's representatives.
A few months ago, my wife and I planned a trip to France. Our outbound itinerary involved taking Amtrak to Boston and then a flight out from Logan. When it became evident that the train was severely delayed, I talked to the station agent; she was sympathetic to our plight, and cancelled our Amtrak tickets for a full refund. We drove two hours to the airport instead (spending a couple hundred dollars on airport parking, which was what we were trying to avoid by taking the train.)
But there are a lot of caveats to this advice, and it might not be applicable to all situations:
We actually had an alternate method of transportation to get us there. The station agent suggested Uber instead, which also might have worked. (In the event, we had never used it before and weren't about to try it out for the first time in a high-stress situation.)
We didn't actually use Amtrak's services at all; if we had taken the delayed train and then tried to claim compensation, we almost certainly would have been out of luck.
The station agent didn't have to give us a refund; the Amtrak tickets we had purchased were non-refundable, so the best we could have expected was a voucher for future travel. I suspect that if our destination wasn't quite so romantic or exotic, the agent might not have been as willing to bend the rules for us.
We didn't try to claim any compensation from Amtrak for the incidental expense of the long-term parking at the airport, or for that matter the gas used to get us to the airport. I'm sure that they would have politely told us to go pound sand if we tried.
Many Travel Insurance policies cover the cost of flights missed because of documentable delays in transfers to the airport, but usually only if they are on a "public conveyance" (as I recall the wording from one policy) and (obviously) if the scheduled time would have permitted catching the flight properly (which probably means the full however-many hours the airline recommends). So a bus would count, and probably a taxi, but not your in-laws promising to drive you and never showing up.
I think that the availability of travel insurance is pretty much your only hope in this situation. Otherwise, the risk is yours to take.
The only way I've ever seen Travel Company A take responsibility for Travel Company B is if A booked B for you (i.e. was part of a package deal). Cruise lines, for instance, will book shore excursions for you or you can book your own, but with the understanding that, if you book it yourself, you're on the hook if you miss the boat (slightly oversimplified for the sake of example). As Gerrit also mentioned, sometimes public transits in Europe will also have agreements for liability, but only in limited circumstances (i.e. they have a partnership with said company). Such agreements are uncommon.
So, no, you cannot compel a bus company to take responsibility for you missing a flight. The only exception here might be something catastrophic (i.e. the bus flips over, catches fire, etc). In that case, you can sue for damages after the fact, and throw the missed flight into that number.