Consistent shipping allows for a simpler shopping cart. If your products are similar weight, size, volume, and density - then the shipping calculation is simpler. The shopping cart software is simpler and cheaper.
For example: a t-shirt fits in the same size box, weighs the same, and costs the same to ship regardless of the size, color, or design on it. On the other hand, if the products you sell are always different: a small, intricate jewelry box, a sturdy oak chair, and a tall, delicate lamp stand with shade - then the shipping is always different. There may be custom box or packaging requirements.
It is a feasible business model to have products with all those different shipping attributes, it just requires a custom shipping quote for every sale. Just build the custom shipping quote into your business model. There needs to be a feedback process to get the quote to the customer and then confirm the sale. Some shopping carts have the UPS Point-to-Point Shipping Calculator installed, which means the software does the shipping calculation for each product during checkout.