Providing an accurate measurement of how long it would take for death to occur after cessation of insulin in an individual who intrinsically makes no insulin is not feasible. This is because such a medical research study would be ethically unacceptable. Any researcher would be obligated to provide insulin.
Natural History of Lack of Insulin
However, you are absolutely correct that individuals with type 1 diabetes cannot simply "take better care of themselves" to avoid their insulin requirement. Autoimmune destruction of the insulin-producing beta cells of the pancreas prevent the individual from making endogenous insulin. As Yukdin (2000. PMID 10752719) notes:
Three quarters of a century after its discovery, insulin is not routinely available in many parts of the developing world. ... In consequence, the life expectancy of a child with newly diagnosed type 1 diabetes in much of sub-Saharan Africa may be as short as 1 year.
Thus, even the early "honeymoon phase", lack of access to insulin is routinely fatal.
Residual Insulin Secretion Decreases Over Time
In reality, however, individuals with type 1 diabetes come to clinical attention once pancreatic insulin secretion reaches a critical non-zero threshold. As Steele and colleagues (2004. PMID 14747294) note:
With time, all individuals with type 1 diabetes are thought to lose insulin production.
Here is a graph from their paper showing decreased insulin production following diagnosis.
Figure 2 from (Steele et al 2004. PMID 14747294) available here.
Thus, how long an individual could survive is likely also correlated to their residual insulin secretion which may be present many years after their diagnosis.
Interestingly, a randomized controlled trial of exercise intervention (one type of "taking care of yourself") among newly diagnosed patients with type 1 diabetes found no significant effect on residual insulin production (Narendran et al 2017. PMID 28905421).
Results of Abrupt Cessation of Insulin
However, an interesting study by Keller and colleagues (1993. PMID 8421367) does provide us a window into what the natural history might look like. In their study, they reviewed cases of prisoners in the New York City prison system who were not provided access to their life-saving insulin therapy. Here is an excerpt of a table from the study:
||Mean ± SD (Range)
|Number of participants
|Number of days since insulin
||3.4 ± 3.3 (1-21)
|Serum glucose (mmol/L)
||27.5 ± 10.0 (11.9-47.2)
|Serum glucose (mg/dL)
||495 ± 180 (215-850)
|Serum bicarbonate (mmol/L)
||14.4 ± 6.2 (3.1-30.0)
|Mean arterial pH
||7.28 ± 0.10 (6.99-7.47)
As you can see, in an average of just 3 and a half days, the prisoners were ill enough for hospitalization, many of them critically.
Cause of Death in Untreated Type 1 Diabetes
The life threatening complications of untreated diabetes are diabetic ketoacidosis and hyperosmolar hyperglycemic state. These conditions result in hyperglycemia, ketone production, elevated plasma osmolality, sodium and potassium derangements, and proinflammatory state. With modern therapy, the ultimate cause of death is most frequently cerebral edema (Lizzo et al 2021. NBK 560723). However, historically, vascular thrombosis and hypovolemic shock were also felt to be important causes of mortality (Clements and Vourganti 1978. PMID 102504)
Overall, patients with type 1 diabetes and no endogenous insulin production definitively require insulin therapy to sustain life. They will require hospitalization in approximately 3 and a half days on average. How long they might survive after the onset of diabetic ketoacidosis or hyperosmolar hyperglycemic state is difficult to determine and likely varies due to the variety of underlying pathologies which might result in mortality and potentially the degree of residual insulin secretion.