A technical correction. In Mandarin, shí is pronounced very close to the sound of "shi" in ship and "shee" in sheep. You are right that it sounds close to "sure" when it is pronounced by people with strong Beijing dialect.
Um. Modern Standard Mandarin pronunciation is based on Beijing dialect, and it doesn't take a Beijing accent to pronounce it like "sure", that's totally standard. Beijing dialect does feature more erhua than other dialects, but even 哪儿 is considered standard, not specific to Beijing dialect. There are definitely dialects (mostly southern iirc) that pronounce shi as in "ship", but that's not Standard Mandarin. Every television show I've seen, for example, pronounces it like "sure".
Edit: To add a clear example, look at 事. Standard Mandarin would pronounce it "sure", Beijing dialect goes even further and pronounces it more like "shar".
I would argue 事儿, the combination of two characters, are pronounced like "sure", in most Northern dialects. (Speaking strictly, some are dialects and some are accents. Let's skip this difference here since it doesn't matter to the current topic.)
Not all the people in television shows, including talk shows, speak Standard Mandarin. This is especially true in shows produced in the north. They adopt Beijing accent to certain degrees, which, more or less, leads to the erhua pronunciation.
For Standard Mandarin, listen to 新闻联播,the official news program from CCTV.
I notice that for non-native Chinese speakers whose mother tongue is Indo-European language, it sounds erhua pronunciation is easier for them than Standard Mandarin. If erhua pronunciation is pushed to extreme, it is called 大舌头。
Also erhua pronunciation is usually NOT used in very formal conditions, like presentation, etc. Some people regards erhua pronunciation as vulgar, except for very widely adopted cases.
I've listened to 新闻联播, the way they pronounce shi sounds much closer to sure than to ship. It is somewhere in-between, and that vowel is more rhotic in Beijing dialect, but Standard Mandarin definitely still rhotacizes that vowel (especially compared to southern dialects where it isn't rhotacized at all).