(This is an extension of my thoughts here)
From the available biographical evidence, we have every reason to believe that Indiana Jones is a good archaeologist. I've previously covered what training Indiana Jones would have received, what professors he would have had, and who his colleagues might have been. I've also looked into what kind of school he taught at
However, not much more can really be said about Indiana Jones' archaeological methodology, because we rarely see him in anything but extraordinary circumstances. The majority of the films are about him acquiring precious or powerful artifacts before someone nefarious does:
The only instance where we see Dr. Jones doing any kind of research or, really, any kind of search that he instigated himself, is at the start of Raiders. The opening shots clearly show a well-equipped team loaded with all kinds of equipment for a proper excavation. It's not long before Indy realizes something is up. He takes Satipo alone into the Hovitos temple, and proceeds through it with extreme caution, showing the requisite skills for any archaeologist: keen observation, knowledge of the cultural context, and careful steps to avoid damaging the architecture or himself. Fully aware that there are less scrupulous people after him, Indy is very much in the right to take the idol and leave. Soon he is betrayed and captured, and we're left without knowing anything else of Indy's own interests.
I'll admit that Indy's trek into the rat-infested tomb under Venice shows him in a more negative light- he rips some burial cloth for a torch. But when he and Elsa reach the knight's sarcophagus, Indy does a non-destructive rubbing of the lid, which, given the time constraints and non-feasibility of photography, seems reasonable.
Indiana Jones is not a tomb raider because we never see him to set out to raid tombs. He clearly meant to do much more at the Hovitos temple than was actually done. He takes the Ark of the Covenant, or attempts to, so that the Nazis can't access the full power of God. He steals the Sankara stone back from the murderous cult that stole it from its proper owners. The entire point of the Last Crusade is that Indy is not interested in the goose chase that his father has gotten caught up in, but in spite of this must take a literal leap of faith to save his father and prevent the Nazis from getting the key to immortality. If the question is tomb raiding or all-powerful Nazis, the correct answer is tomb raiding.
If he's not a tomb raider, then what is he?
The Indiana Jones films are explicitly modeled after the pulp magazines, radio shows, and early comic books of the 1930s-1940s, as well as the adventure novels of the preceding decades. There's a clear inspiration for Indy in Allan Quartermain, Professor Challenger, and Doc Savage: characters whose bravado or inquiry lead to run-ins with exotic tribes, Lost Worlds, and ancient legendary ruins. This is the same tradition that gave us the secret identity: The Spider was >!Richard Wentworth!<, The Shadow was >!Kent Allard!<, and Batman was >!Bruce Wayne!<. In a literary sense, Indiana Jones falls right into this trope. He's a vigilante who hides his activities from the public, but is known by an elite few for his exploits and by a trusted team of supporting allies.
For this reason, I would argue that Indiana Jones is an archaeologist in the same way that Hal Jordan is an Air Force pilot or Peter Parker is a news photographer. Hal Jordan's pilot skills help him as a Green Lantern just as Indiana Jones' archaeology skills help him as an adventurer. The stories we see about him don't show him in his civilian career, but in pulp hero mode.
But what of the archaeology of it?
What Dr. Jones is doing is what we would call surface collection or salvage archaeology. Before any excavation, and sometimes during the initial survey and registration of a site, a team will collect artifacts that can be found on the surface, This will be done systematically and unobtrusively. Some archaeologists will plot transects (i.e. parallel, regularly spaced lines) that a team will walk across to record features, identify locations of high artifact concentration, and, as necessary, collect surface artifacts. Other projects prefer to this in grids, plotting out regular boxes that can be then assigned values. In either case, should artifacts be collected, their original location will be noted on a hand drawn map or with a GPS point, and they will often be photographed in situ.
Why would archaeologists do this?
First of all, because these artifacts are on the surface, it is unlikely that they are what we call a "primary deposition," that is, where the culture that used it left. It is likely a case of secondary deposition, decontextualized by burrowing animals, erosion, or agricultural activity. Thus, we only lose so much information by collecting it. There is an entire disciple called "taphonomy" that is dedicated to the study of how things end up where they do, but, even then, the amount of taphonomic information on the surface is limited.
Second, surface artifacts are very helpful for understanding a site before excavation, and therefore directing where further research takes place. Excavation is costly and destructive, and should be done as little as possible. Archaeologists want to be sure that any excavation they do will offer useful data. The form of artifacts can inform what kind of activities were happening in area, their style can help date the site, and their area of distribution can help define the coverage of habitation.
Lastly, surface collection can be necessary in areas where looting is a real and present danger. An archaeological dig is a beacon for looters. In my own work, I've returned to an operation in the morning and found shovels and stacks of ceramic sherds left by lazy looters in nearby tombs. When artifacts are sitting in the open and looters are known to be active, it is much better to extensively document those artifacts and collect them than it is for them to be looted and never seen again.
Thus, in the situation where a golden idol is sitting on a pedestal at a site you're about to excavate with a series of spikey traps in front of it, you grab that idol.
I mentioned salvage archaeology because that's the closest real-world correlate to getting the Ark of the Covenant before the Nazis do. There's an entire commercial field of archaeology called contract archaeology or Cultural Resource Management (CRM). In many countries, before major construction can begin, a contractor must submit an impact statement showing that the construction will not have a substantial environmental or geological impact. They must also prove that it won't have detrimental archaeological impact. If the state is building a highway bypass, or an electrical company is building a new line, or a town wants to institute a new technological park, they will hire a CRM team to survey the land that will be built on. We go out, dig a series of test pits, and look for any evidence of occupation. This is quick and dirty work. On my best days I could dig 25 of these "Phase 1" test pits. Unlike the slow, careful survey I describe above, Phase 1 CRM surveys try to cover as much ground as possible as quickly, but as orderly, as possible. You dig a hole (something like 30cm x 30cm, and down until you hit bedrock or sterile clay), you record the soil strata, and mark if you found any cultural material. Then you move on. When a hole is positive, you might dig some more in a grid around it. If there's a large cluster of positive test pits, you might move on to Phase 2: carefully excavating 1m x 1m test squares to identify cultural features (e.g. post holes, trash pits). If you've got cultural features, a Phase 3 excavation might be begin- you've got a site!
You've also (usually) got a mad construction company. One week you're excavating is one week they're not building and one week their client has to wait. The fact that any archaeological survey is happening is great. But the pressure is always there to get moving so other folks can get building. There are great CRM firms, and I was privileged to work for an incredible one. There are also terrible ones. But in both cases, you are operating with the knowledge that everything you see is about to be destroyed to make way for a road, building, or parking lot. That means that if we do a Phase 2 operation and don't find any cultural features, we go in with a bulldozer to scrape away huge swaths of dirt 10cm at a time. Is this destructive? Heckin' duh. But it's exactly what's going to happen to the land anyways. It's a last ditch effort to find and salvage any cultural material that may be there.
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A side question on the history of archeology, would Dr Jones have been a good or bad Archeologist in his period?