NEW YORK — The city’s homeless shelter population has hit an all-time high as hundreds of Latin American migrants continue to pour into New York every week as part of a crisis that’s driving the local social safety net to the brink of collapse. The previous record — 61,415 individuals in city shelters on Jan. 12, 2019 — was first cracked over the weekend, data from the Department of Homeless Services show. On Monday, the latest day for which data is available, the tally reached 62,174. In addition to setting a new population record, the average length of stay has also surged to all-time highs,…