On the first point: Troy didn’t disappear at the end of the Bronze Age. It was a major city from the late 700s BCE to around 500 CE. Around 500 it was hit by a major earthquake, and it gradually faded away over the subsequent centuries. There may still have been people living there when the Ottomans occupied the area in the early 1300s, but maybe not. Throughout all of antiquity, when people referred to Ilium (its strict name), they were talking about a living contemporary city. And for the most part they were pretty happy to identify it with ‘Homer’s Ilium’ in a loose sense, in the same way that when you think about the setting of Robin Hood, you’re happy to slide between the real Nottingham and the one in the legend.

Classical Ilium is extremely copiously attested in ancient sources, and it was a major tourist site: the attractions ranged all over the Troad, from the supposed graves of Achilles and Patroclus at Sigeium, to Ilium itself, to Alexandria Troas on the coast to the south. After the mediaeval period, when western Europeans started thinking of themselves as the heirs of the classical Greco-Roman world, still they didn’t visited the area much up until the 1600s-1700s. Westerners were perfectly well aware of the general location of Ilium, but they didn’t know the exact location of the city limits, so to speak. So we find people like George Sandys and Robert Wood visiting and writing about the place, and complaining because they can’t work out where things are supposed to be.
The precise location of the citadel of classical Ilium was determined by Edward Daniel Clarke in 1801, and confirmed by excavations conducted by John Brunton in the 1850s and Frank Calvert in 1865.

What happened in the 1870s was (a) the finding of the pre-classical, Bronze Age, city, beneath classical Ilium; and (b) the settling of an argument over whether Ilium had moved around.

See, a 1st century geographical writer, Strabo, reported a tradition that ‘ancient Ilium’ hadn’t been in the same place as the contemporary city. He accepted a variant, according to which ‘ancient Ilium’ had been 6 km to the ESE of the contemporary city. In 1785 a French diplomatic aide, Jean-Baptiste Le Chevalier, visited the area armed with a copy of Strabo and decided that Strabo had been right (though paradoxically he also believed Strabo knew nothing about the topography). Le Chevalier misidentified the location of the classical city, 5 km to the NW of its actual location, and he put ‘Homer’s Ilium’ at a village called Pınarbaşı (‘Bunarbashi’), 14 km to the SSE of his supposed classical Ilium.

A lot of people accepted Le Chevalier’s argument, even though his surveying was not very expert, and even though he relied far more on Homer than on physical evidence. The correct location for the classical city was identified in 1801, as I mentioned. The result was that early 1800s maps put ‘New Ilium’ (Ilium Novum) at the correct location, but ‘Old Ilium’ (Ilium Vetus) at this random village away to the south. Le Chevalier’s argument was firmly rebutted by Charles Maclaren in 1822, but it took a while to die. An Austrian diplomat excavated at Pınarbaşı in 1864, and found nothing at all.

When Schliemann arrived in the late 1860s, he was planning to dig at Pınarbaşı too. Calvert persuaded him otherwise. The people who still live in Pınarbaşı today should make an annual dua for Calvert, or something, because if not for him Schliemann would probably have blown the entire village to pieces.
Schliemann ended up digging where he didn’t want to dig, without waiting for a government permit, blowing the hill sky-high with dynamite, destroying hundreds of tons of archaeology and millennia of evidence, in his pursuit of anything that might perhaps look like it might match Homer. And eventually he did find something he didn’t want to destroy: Troy II, as it’s now known, dating to the late 2000s BCE. (He went straight through the archaeology of the 1000s BCE. It’s all irretrievably destroyed.)

He did prove something: he proved that Strabo and Le Chevalier were wrong. Most people had already accepted that by the 1870s; and Ilium itself never needed discovering. So I’d put an awful lot of quotation marks around any use of the word ‘discover’ in connection with Schliemann.

• On the second point, I have less to say, because I don’t know a huge amount about the public reaction to Schliemann’s publications. Certainly Schliemann portrayed himself as discovering Troy. He relied on conflating three things:
• the Bronze Age city that he had excavated
• classical Ilium as it existed at the time when Homer might imaginably have visited
• the city depicted in the Iliad

Strabo’s preferred site was a village ca. 5.5 km away (30 stadia), in the direction of Mount Ida/Kaz, that is to say, at a bearing of 119° from classical Ilium. This would imply a spot near the E87 road, about 3 km north of Taştepe. Le Chevalier got the site of classical Ilium wrong — he put it much closer to the coast, 5 km northwest of the actual site — and he put Strabo’s ‘ancient Ilium’ at Pınarbaşı, nearly 14 km from where he thought classical Ilium was.
Timeline: Hisarlık vs. Pınarbaşı
ca. 20. Strabo, Geography 13.1.35, reports a claim that ‘ancient Ilium’ was 30 stadia inland from the classical city.
Richard Pococke states that Troy was buried underneath classical Ilium.
Robert Wood, after visiting the Troad in 1750, notes that Homer’s descriptions don’t match the real topography.
Jean-Baptiste Le Chevalier, after examining the area in 1785, argues that Homer’s Troy was at a hill on the south side of the village of ‘Bunarbashi’, and that classical Ilium was just a stone’s throw from the coast, at the mouth of the Scamander (modern Menderes). He accepts Strabo’s idea that the site had moved over time, but rejects most of Strabo’s details, in the belief that he had a poor understanding of the topography.
Jacob Bryant criticises Le Chevalier’s argument, mainly using Homeric references. The argument continues in subsequent years, with Le Chevalier backed up by his former boss, Choiseul-Gouffier.
Edward Daniel Clarke visits the Troad and correctly identifies Hisarlık as (part of) classical Ilium.
Charles Maclaren argues that both ‘the Ilium of Homer’ and classical Ilium were at Hisarlık.
P. W. Forchhammer writes a survey labellng Hisarlık as ‘New Ilium’ (Ilium Novum), Pınarbaşı as ‘Troy, or Old Ilium’ (Troja vel Ilium Vetus), and labels Strabo’s preferred site separately in the hills south of Dümrek.
1850s. John Brunton excavates at Hisarlık and finds a column capital and a Roman mosaic.
Frank Calvert, after investigating Hisarlık and Pınarbaşı, opts for Hisarlık.
Johann Georg von Hahn, an Austrian consul, excavates at Pınarbaşı and finds ‘nothing but the natural soil.’
Calvert excavates at Hisarlık and finds parts of a temple and a wall.
Heinrich Schliemann arrives in the Troad and investigates Pınarbaşı, until Calvert persuades him that Hisarlık is the correct spot.
1871, Schliemann starts uncovering evidence of a Bronze Age city under the classical city, proving Strabo (and Le Chevalier) wrong.