I’m sorry to say that the letting agents you have spoken to are telling the truth.
When a flat is advertised, all the viewing slots will be filled within a few hours . Viewings will probably be the next day - certainly within a few days and many viewers will submit their application form that same day or the next day. If your application is chosen and you pass the vetting, you will need to sign the contract and pay the deposit within a week.
Most reputable agencies won’t let hundreds of people view a flat - probably just 15-20. There’s no point if they know it’s a good property and 1/3 or 1/2 of viewers will apply.
You need to be able to view flats within say 72 hours of the advert going up. So either you come up for a week and try to get something or you have someone view them for you ( if you have anyone ) .
I’d not recommend renting a flat without viewing, especially as you don’t know the area. Anyway most agents won’t let you do this as there’s a high chance you will move out quickly.
Rent levels - again you’ve had accurate information. If you don’t meet their affordability criteria you will need a guarantor . Landlords can’t afford to take tenants who may not pay the rent, as government Covid rules mean it’s very hard to get them out. They can end up paying the mortgage / insurance / factors bills / legal bill to end up with a trashed property a year down the line.
Again, most landlords won’t take people who want to pay 3 -6 months up front because they don’t meet the affordability criteria, as often they just stop paying after that.
I’m afraid you will be quite limited as to the areas of Glasgow you can afford on £600 for a one bed. Posters are right that Shawlands/ Strathbungo are nice - they are great but you will struggle to get a decent one bed in Shawlands for £600.
I think you will need to stay further out at first, at least until you get back to work after maternity leave and you have a higher household income.
I’m sorry I can’t give you more cheerful news.
What Scottish MNers CAN do is tell you if a letting agency is well known here and also what various areas are like.
And if you tell us where your partner’s new job is we can tell you areas that would be commutable.