Hello, I wrote an answer here about the development (or not) of burhs into urban sites that you might find useful.

Burh is a complex term that has both a highly specific and an incredibly generic use which can slightly confuse issues. While the term is used specifically to refer to the network of late 9th Century Alfredian fortresses and their early 10th Edwardian and Æthelflædian successors, burh as a word just means an enclosure, so it's also widely used generically to refer to any enclosed site, including cattle pens and sheep folds, pre-9th century fortresses and repurposed hillforts. What this means in practical terms is that burh toponyms - bury, berry, bry, etc. - are actually largely absent from actual burghal fortress sites, but are incredibly common in other contexts. Actually, in relation to the actual burghal network (which I talked about the fuction of here) burh toponyms are more commonly related to local defensive strongholds than they do the burhs themselves. Warburton in Cheshire, for example is derived from weard-burh-tun or "defensive-enclosed-town", and Burwardsley (burh-weards-hlaw or "enclosed defensive hill") is a fortified hilltop between the burhs at Chester and Eddisbury. Eddisbury is one of the few burhs with a burh toponym, either named for the ealdorman or thegn entrusted with its defence, or perhaps the previous fortification on the site.

We actually have an extensive and specific record of which sites were "officially" burhs - the Alfredian sites are listed in a late 9th/early 10th century document known as The Burghal Hidage and the subsequent Edwardian and Æthelflædian sites are detailed in the Anglo-Saxon Chronicle and its contemporary The Mercian Register. Mercian burhs included Chester, Stafford, Tamworth, Bridgnorth, Chirbury and Eddisbury, and are particularly useful for identifying what differentiates a burh from other sites. Burhs specifically functioned as garrison sites and communications hubs designed to control major transport nexi, thwart Danish tactics of rapid manoeuvre, and use main transport routes to rapidly respond to threats. In many cases, particularly in Wessex, these naturally coincided with the locations that lend themselves to major settlements. Maintaining a burh, in terms of fortification and manpower, could be very expensive, so a location at a pre-existing settlement could be a great boon, but the state also wanted the minimum number of burhs it could strategically afford, so sites were typically around 20-40 miles apart. This meant that, while major settlements were preferred they could be skipped if they weren't strategically necessary, and burhs could be built in new locations if this was necessary. The key factor is that burhs have an intrinsically military purpose that occasionally coincides with the locations of major settlements

London, for example, was already a prosperous trading settlement thanks to its location at the confluence of a number of Roman roads and its site on the Thames as both a crossing point and a site enabling both maritime and riverine trade. This context made it strategically important as a burh as well. In the case of Tamworth, it was a little close to Stafford, but was likely chosen as a burh for the propaganda reasons of its association as Offa's Mercian capital. The burhs at Bridgnorth, Chirbury and Eddisbury, however, were at relatively sparsely-populated but strategically vital locations that required a garrison site. Chirbury exercised control control over the Welsh frontier but is just a small village today. The burh at Bridgnorth enabled the growth of nearby settlements in Shropshire and guarded a vital river crossing, but was itself abandoned by 1066, with the modern town dating from a 12th century castle re-established on the site. Eddisbury was an extensively refortified Iron Age hillfort, garrisonned to provide protection to the Cheshire Plain and encourage the rapid proliferation of prosperous settlement there in the 910s, but abandoned likely as early as the 920s once the Danelaw frontier had been recaptured and the Viking risk had passed, as keeping a substantial garrison in an otherwise unproductive hilltop was a major expense.

Although primarily military sites, the expansion of Cerdicing royal authority meant that they were also intended as civic centres with a proto-urban function. It was hoped that as sites of trade, law, minting and religion, this process would allow the crown to collect more extensive tax revenues, be more involved in legal oversight, and exercise greater administrative control. In pre-existing settlements, this made sense as the infrastructure was already largely present (although this didn't stop the extensive re-modelling of street schemes and the construction of new churches and institutions), and in new sites this also made sense as churches and traders often stood to gain from moving inside the protective aegis of a burh. Of course, these models didn't always work. Shrewsbury, for example, was a major trading town and mint site with significant fortifications, but wasn't in a strategic location to require becoming a burh. However, as a pre-existing civic site, it negated the need for those functions to be established at other Mercian sites. Eddisbury and Runcorn likewise, could be solely military sites as civic functions were absorbed by the major city at Chester. In Dorset, the old county town at Dorchester and the Royal vil at Wimborne/Kingston Lacy also meant that the burh at Wareham (weard-ham, "guard-settlement") didn't need to develop a civil function beyond its pre-existing fishing/trade port. Visa versa, the burh at Wareham meant that the county town at Dorchester didn't need to be developed as a burh. At Wallingford and Stafford, on the other end of the spectrum, the development of the fortifications at major river crossings on Roman roads, the continued presence of large garrisons and the accumulation of attendant religious and civic sites for the use of the garrison, and populations attracted by the safety they offered, meant that those sites continued to develop into urban settlements after their burghal function was no longer necessary.

So in terms of a vague conclusion: as Shakespeare definitely didn't say, some sites are born burhs and others have burhness thrust upon them. Some pre-existing towns were developed into burhs because the conditions for a successful town are usually strategically important (e.g. London, Hamwic, Tamworth). Some major settlements did not become burhs because they were not in strategically important sites despite their wider economic, political and religious importance (e.g. Gloucester, Shrewsbury, Dorchester). Some burhs developed into settlements as the safety they offered and the civil functions they were given remained even as their military importance declined (Chester, Stafford, Wallingford). And some burhs never developed into settlements at all because they were in strategically important but sparsely populated areas or were never intended to have an urban function in the first place (Cyneuit, Eddisbury, Runcorn, Chrirbury).