score:6
As always, any of this information is subject to change on short notice, but:
Egypt offers a special Sinai resorts entry permission to visitors from some countries (including the US) when arriving at the Taba crossing or Sharm-el-Sheikh airport and not traveling beyond the resort zone. Otherwise, an e-visa is available in advance and costs US$25 for a single entry or US$60 for multiple entries.
Note that the US government currently warns against travel to "the Sinai Peninsula (with the exception of travel to Sharm El-Sheikh by air) due to terrorism."
Israel does not offer any special relaxed-visa zone but US citizens may enter without obtaining a visa anyway.
Jordan offers a free "Special Economic Zone" visa to visitors from many countries, including the US, who arrive at the land, air, or sea border in Aqaba. Unlike the Egyptian program, you may travel to other parts of Jordan on this visa after arriving in Aqaba, but there are nonetheless some requirements:
If you do not want to participate in the SEZ visa program, you can obtain a "Jordan Pass" which includes an entry visa and admission to some Jordanian tourist attractions bundled (though you also must stay at least three nights in Jordan), or a regular visa on arrival at the checkpoint. Note that the fine for leaving Jordan through a different border crossing on the SEZ visa is the same (JOD 40) as the cost as the regular visa, so it is okay to take the SEZ visa and then just pay the fee on departure if you are unsure of your plans or your plans change.
Saudi Arabia does not offer any special relaxed-visa zone. US citizens can obtain a Saudi eVisa in advance. It is supposedly possible to obtain a visa on arrival as well. A recent traveler report says that "none of the border guards knew about it and confusion was perfect" though they were eventually allowed to purchase the visa and enter.
It is probably overall easier to travel from one country to the next, rather than basing yourself in one spot and making repeated border crossings to neighboring countries.
(Remember that immigration and travel rules can change at any time.)