you can’t, although i suppose if you were dedicated and had a computer/video editing software you could edit out every frame that had the red recording logo in favor of the unobtrusive one. but that would require you to record for much longer
fliqlo seems to do this but doesn’t provide the date, and your phone has to be unlocked (i.e. it’s not a lock screen widget)
not a solution but a suggestion, when you pull down to see notifications try pulling down from where the clock is displayed on screen - i’ve had a good success rate doing so from there. i can’t speak to speed, but i’m sentimental and i know in my transitional periods between phones i always feel a little reticent to switch fully/i try to come up with excuses to keep the old device instead because i have happier feelings with it (for lack of better terms atm). idk if that’s what’s happening to you here, but i think you should give yourself a day or two where you commit yourself to finding things that your 13 improves on or does now than your 6s, and if you STILL don’t like it, you could return it in favor of a SE 2022 so you at least have the same form factor
an answer you may not like but may work for you is an android phone, both pixels and samsungs have the ability to have separate profiles for work and personal and support multiple users in the event that your company doesn’t have an mdm
based on the US official repair estimate site it looks like it’ll be about $150 USD; idk what country you’re in so you may have to double check